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The mechanics of the contactphase in trampolining
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The Mechanics of the Contact Phase in
Trampolining
by
David John Burke
A Doctoral Thesis
Submitted in partial fulfilment of the requirements for the award of
Doctor of Philosophy of Loughborough University
February 2015
© David John Burke 2015
i
ABSTRACT
The Mechanics of the Contact Phase in Trampolining
David John Burke, Loughborough University, 2015
During the takeoff for a trampoline skill the trampolinist should produce sufficient
vertical velocity and angular momentum to permit the required skill to be
completed in the aerial phase without excessive horizontal travel. The aim of this
study was to investigate the optimum technique to produce forward somersault
rotation. A seven-segment, subject-specific torque-driven computer simulation
model of the takeoff in trampolining was developed in conjunction with a model
of the reaction forces exerted on the trampolinist by the trampoline suspension
system. The ankle, knee, hip, and shoulder joints were torque-driven, with the
metatarsal-phalangeal and elbow joints angle-driven. Kinematic data of
trampolining performances were obtained using a Vicon motion capture system.
Segmental inertia parameters were calculated from anthropometric measurements.
Viscoelastic parameters governing the trampoline were determined by matching
an angle-driven model to the performance data. The torque-driven model was
matched to the performance data by scaling joint torque parameters from the
literature, and varying the activation parameters of the torque generators using a
simulated annealing algorithm technique. The torque-driven model with the scaled
isometric strength was evaluated by matching the performance data. The
evaluation produced close agreement between the simulations and the
performance, with an average difference of 4.4% across three forward rotating
skills. The model was considered able to accurately represent the motion of a
trampolinist in contact with a trampoline and was subsequently used to investigate
optimal performance. Optimisations for maximum jump height for different
somersaulting skills and maximum rotation potential produced increases in jump
height of up to 14% and increases of rotation potential up to 15%. The optimised
technique for rotation potential showed greater shoulder flexion during the recoil
of the trampoline and for jump height showed greater plantar flexion and later and
ii
quicker knee extension before takeoff. Future applications of the model can
include investigations into the sensitivity of the model to changes in initial
conditions, and activation, strength, and trampoline parameters.
Keywords: trampoline, trampolining, takeoff, simulation, model, optimisation,
torque-driven
iii
ACKNOWLEDGMENTS
I would like to thank my supervisors, Prof. Fred Yeadon and Dr. Mike Hiley, for
their guidance and support throughout this project. I am also grateful to the School
of Sport, Exercise, and Health Sciences and British Gymnastics for their financial
support. I would like to thank you my family and friends for all their support
throughout my studies. Finally I would like to thank everyone in the Sports
Biomechanics and Motor Control Research Group and associates (Mark, Matt,
Sam, Paul, Glenn, Ravina, Fearghal, Laura-Anne, Stuart, Dara, Romanda, Ben,
Dimitrios, Emma, Martin, Neale, Monique, Phil, Felix, Michael, Peter, David,
Helen, Akiko, Ciaran, Gheorghe, Mickael, Steph, Lesley, Bob, John, Alec, David,
Sarah, and Jo) who have provided assistance and friendship during my time in
Loughborough.
iv
To Ola, and My Family
v
Contents
ABSTRACT ............................................................................................................. i
ACKNOWLEDGMENTS ..................................................................................... iii
Contents ................................................................................................................... v
List of Figures ........................................................................................................ xi
List of Tables ......................................................................................................... xv
Chapter 1: Introduction ............................................................................................ 1
1.1 Previous Trampolining Research .............................................................. 2
1.2 Statement of Purpose ................................................................................. 3
1.3 Research Questions ................................................................................... 3
1.4 Chapter Organisation ................................................................................. 4
Chapter 2: Review of Literature............................................................................... 6
2.1 Chapter Overview...................................................................................... 6
2.2 Overview of Trampolining Research ........................................................ 6
2.2.1 Takeoff ............................................................................................... 7
2.2.2 Height ................................................................................................. 7
2.2.3 Rotation ............................................................................................ 10
2.2.4 Inter-relationship of Height and Rotation ........................................ 11
2.2.5 Summary .......................................................................................... 12
2.3 The Trampoline ....................................................................................... 12
2.3.1 Modelling the Trampoline ............................................................... 13
2.3.2 Measuring Trampoline Parameters .................................................. 15
2.3.3 Limitations of Previous Trampoline Models ................................... 17
2.4 Computer Simulation Models ................................................................. 17
2.4.1 Overview .......................................................................................... 17
2.4.2 Trampolining Simulation Models .................................................... 18
vi
2.4.3 Non-trampolining Takeoff Simulation Models................................ 20
2.4.4 Wobbling Mass Models ................................................................... 21
2.4.5 Summary .......................................................................................... 23
2.5 Simulation Model Input........................................................................... 24
2.5.1 Strength Parameters ......................................................................... 24
2.5.1.1 Muscle Modelling ..................................................................... 24
2.5.1.2 Strength Measurement .............................................................. 27
2.5.1.3 Summary ................................................................................... 30
2.5.2 Body Segmental Inertia Parameters ................................................. 30
2.5.3 Kinematic Data ................................................................................ 32
2.5.3.1 Image Analysis ......................................................................... 32
2.5.3.2 Data Processing ........................................................................ 33
2.6 Optimisation ............................................................................................ 34
2.7 Chapter Summary .................................................................................... 35
Chapter 3: Data Collection ..................................................................................... 37
3.1 Chapter Overview.................................................................................... 37
3.2 Kinematic Data Collection ...................................................................... 37
3.2.1 Camera Set-up .................................................................................. 37
3.2.2 Data Collection ................................................................................ 38
3.2.3 Body Segmental Inertia Parameters ................................................. 41
3.3 Kinematic Data Processing ..................................................................... 44
3.3.1 Joint Centre Positions....................................................................... 44
3.3.2 Joint Angle Time Histories .............................................................. 47
3.3.3 Trampoline Bed Movement ............................................................. 51
3.4 Chapter Summary .................................................................................... 52
Chapter 4: Model Development ............................................................................. 53
4.1 Chapter Overview.................................................................................... 53
vii
4.2 The Trampolinist Model .......................................................................... 53
4.3 Foot/Suspension-System Interface .......................................................... 54
4.3.1 Kinetics of the Foot/Suspension-System Interactions ..................... 55
4.3.2 Locating the Centre of Pressure during Foot Contact ...................... 59
4.3.3 Determining the Natural Slope of the Surface of the Suspension
System during Depression .............................................................................. 62
4.3.4 Equations of Motion......................................................................... 63
4.4 Chapter Summary .................................................................................... 64
Chapter 5: The Angle-Driven Model ..................................................................... 65
5.1 Chapter Overview.................................................................................... 65
5.2 Visco-Elastic Parameter Determination .................................................. 65
5.2.1 Model Inputs .................................................................................... 66
5.2.2 Cost Function ................................................................................... 67
5.2.3 Results .............................................................................................. 68
5.3 Evaluation of the Angle-Driven Model ................................................... 71
5.4 Chapter Summary .................................................................................... 73
Chapter 6: The Torque-Driven Model ................................................................... 74
6.1 Chapter Overview.................................................................................... 74
6.2 Structure of the Torque-Driven Model .................................................... 74
6.3 Torque Generators ................................................................................... 76
6.3.1 Torque – Angle – Angular Velocity Relationship ........................... 77
6.3.1.1 Torque – Velocity Relationship ................................................ 77
6.3.1.2 Differential Activation .............................................................. 79
6.3.1.3 Torque – Angle Relationship .................................................... 80
6.3.2 Strength Parameters ......................................................................... 81
6.3.3 Muscle Activation Profiles ............................................................... 82
6.3.4 Strength Parameter Scaling Factors ................................................. 84
viii
6.3.5 Metatarsal-Phalangeal Joint ............................................................. 85
6.4 Model Evaluation .................................................................................... 86
6.4.1 Model Input ...................................................................................... 87
6.4.2 Model Variables ............................................................................... 88
6.4.3 Cost Function ................................................................................... 90
6.4.4 Strength Scaling Matching Results .................................................. 93
6.4.4.1 Joint Angles .............................................................................. 96
6.4.4.2 Joint Torque Activation Profiles ............................................... 99
6.4.4.3 Joint Torques .......................................................................... 101
6.4.4.4 Strength Scaling Factors ......................................................... 103
6.4.5 Fixed Strength Matching Results ................................................... 104
6.4.5.1 Joint Angles ............................................................................ 106
6.4.5.2 Joint Torque Activation Profiles ............................................. 108
6.4.5.3 Joint Torques .......................................................................... 110
6.4.6 Discussion ...................................................................................... 112
6.4.6.1 Joint Angles ............................................................................ 112
6.4.6.2 Movement Outcomes at Takeoff ............................................ 112
6.4.6.3 Summary ................................................................................. 113
6.5 Chapter Summary .................................................................................. 113
Chapter 7: Optimisation and Applications ........................................................... 114
7.1 Chapter Overview.................................................................................. 114
7.2 Application to Research Questions ....................................................... 114
7.3 Calculation of Kinematic Variables ...................................................... 115
7.3.1 Jump Height ................................................................................... 115
7.3.2 Flight Time ..................................................................................... 115
7.3.3 Rotation Potential ........................................................................... 116
7.4 Optimisation Method ............................................................................. 117
ix
7.4.1 Maximum Height Method .............................................................. 117
7.4.2 Maximum Rotation Method ........................................................... 118
7.5 Optimisation Results ............................................................................. 119
7.5.1 Maximum Jump Height Results ..................................................... 119
7.5.1.1 Joint Angles ............................................................................ 123
7.5.1.2 Joint Torque Activation Profiles ............................................. 126
7.5.1.3 Joint Torques .......................................................................... 130
7.5.1.4 Summary ................................................................................. 132
7.5.2 Maximum Rotation Results ........................................................... 132
7.5.2.1 Joint Angles ............................................................................ 135
7.5.2.2 Joint Torque Activation Profiles ............................................. 138
7.5.2.3 Joint Torques .......................................................................... 141
7.5.2.4 Summary ................................................................................. 143
7.6 Discussion ............................................................................................. 143
7.7 Chapter Summary .................................................................................. 144
Chapter 8: Discussion and Conclusions ............................................................... 145
8.1 Chapter Overview.................................................................................. 145
8.2 Research Questions ............................................................................... 145
8.3 Discussion ............................................................................................. 147
8.3.1 Simulation Model ........................................................................... 147
8.3.2 Foot-Suspension System Interface ................................................. 148
8.3.3 Performance Data ........................................................................... 148
8.3.4 Anthropometric Data...................................................................... 149
8.3.5 Kinematic Data Processing ............................................................ 149
8.3.6 Determination of Suspension System Interface Parameters .......... 150
8.3.7 Torque Parameters ......................................................................... 150
8.3.8 Evaluation of the Torque-driven Model......................................... 151
x
8.3.9 Cost Scores and Penalties .............................................................. 151
8.4 Future Research ..................................................................................... 152
8.4.1 Robustness ..................................................................................... 152
8.4.2 Initial Conditions............................................................................ 153
8.4.3 Subject-Specific Parameters .......................................................... 153
8.4.4 Sensitivity to Model Parameters .................................................... 153
8.4.5 Application to Different Trampoline Contacts .............................. 154
8.5 Conclusions ........................................................................................... 154
References ............................................................................................................ 155
APPENDICES ..................................................................................................... 168
APPENDIX 1: Informed Consent Form .......................................................... 169
APPENDIX 2: Anthropometric Measurements ............................................... 174
APPENDIX 3: AutolevTM
4.1 Command Files ................................................ 177
APPENDIX 4: Joint Torque Profiles from Angle-Driven Matching Simulations
.......................................................................................................................... 191
APPENDIX 5: Torque Generator Activation Parameters from Strength Scaling
Torque-Driven Matching Simulations.............................................................. 194
APPENDIX 6: Torque Generator Activation Parameters from Fixed Strength
Torque-Driven Matching Simulations.............................................................. 197
APPENDIX 7: Torque Generator Activation Parameters from Optimisations 200
xi
List of Figures
Figure 3.2.1 A view of the experimental set up. .................................................... 38
Figure 3.2.2 Illustration of the 16 camera positions for data collection. ............... 38
Figure 3.2.3 Photograph showing the marker set employed to collect position data.
................................................................................................................................ 39
Figure 3.2.4 The foot modelled by a triangle and rod segment. ............................ 43
Figure 3.3.1 The transposition of foot markers. ..................................................... 44
Figure 3.3.2 Representation of the captured motion of the trampolinist during (a)
straight bouncing, (b) straight front somersault, (c) 1¾ (open) piked front
somersault, (d) 2¾ tucked front somersault, (e) triffus piked (before initiating
twist)....................................................................................................................... 48
Figure 3.3.3 Representation of the captured motion of the trampolinist during (a)
piked back somersault, (b) straight back somersault, (c) 1¼ straight back
somersault, (d) tucked double back somersault, (e) piked double back somersault.
................................................................................................................................ 49
Figure 3.3.4 Joint angle time histories of the knee (solid), hip (dashed), and
shoulder (dotted) during the contact phase directly preceding different
somersaults. ............................................................................................................ 50
Figure 3.3.5 Plot of normalised bed depression against normalised contact time for
15 trials. .................................................................................................................. 51
Figure 4.2.1 A seven-segment angle-driven model of a trampolinist. ................... 54
Figure 4.3.2 Free body diagram showing the vertical forces acting on the
trampolinist, G, and trampoline, B. ........................................................................ 55
Figure 4.3.3 Graph showing the development of impact force as a function of T.57
Figure 4.3.4 Showing the three separate reaction forces acting on the foot. ......... 60
Figure 4.3.5 The points used to determine the natural slope of the suspension
system..................................................................................................................... 62
Figure 4.3.6 Graph showing the relationship between natural slope and the
depression of the surface of the suspension system. .............................................. 63
Figure 5.3.1 Comparison of F4 trampoline contact phase between recorded
performance (above) and simulated performance (below). ................................... 72
xii
Figure 5.3.2 Comparison of B4 trampoline contact phase between recorded
performance (above) and simulated performance (below). ................................... 73
Figure 6.2.1 A seven-segment torque-driven model of a trampolinist. ................. 75
Figure 6.3.1 The muscle tendon complex consisting of a contractile component
and series elastic component for both (a) a joint extensor and (b) a joint flexor. .. 76
Figure 6.3.2 The four-parameter maximum torque function comprising branches
of two rectangular hyperbolas with asymptotes T = -Tc and ω = -ωc; and T =
Tmax and ω = ωe (Yeadon et al., 2006). ................................................................ 79
Figure 6.3.3 The three parameter differential activation function in which the
activation a rises from amin to amax with a point of inflection at ω = ω1 (Forrester et
al., 2011). ............................................................................................................... 80
Figure 6.3.4 Example of a muscle activation profile. ............................................ 83
Figure 6.4.1 Comparison of F1 trampoline contact phase performance (above) and
torque-driven simulation (below)........................................................................... 95
Figure 6.4.2 Comparison of F3 trampoline contact phase performance (above) and
torque-driven simulation (below)........................................................................... 96
Figure 6.4.3 Comparison of the joint angles during matched simulations (solid
line) and performance (dashed line) of F1 (left) and F3 (right). ............................. 98
Figure 6.4.4 Activation time histories of the extensors (solid line) and flexors
(dashed line) of F1 (left) and F3 (right). ............................................................... 100
Figure 6.4.5 Comparison of the joint torques during matched torque-driven
simulations of F1 (left) and F3 (right). ................................................................. 102
Figure 6.4.6 Comparison of F1 trampoline contact phase performance (above) and
fixed strength torque-driven simulation (below). ................................................ 105
Figure 6.4.7 Comparison of F3 trampoline contact phase performance (above) and
fixed strength torque-driven simulation (below). ................................................ 106
Figure 6.4.8 Comparison of the joint angles during fixed strength matched
simulations (solid line) and performance (dashed line) of F1 (left) and F3 (right).
.............................................................................................................................. 107
Figure 6.4.9 Matched activation time histories of the extensors (solid line) and
flexors (dashed line) of F1 (left) and F3 (right) from the fixed strength protocol.
.............................................................................................................................. 109
xiii
Figure 6.4.10 Comparison of the joint torques during fixed strength matched
torque-driven simulations (solid line) and strength scaling matched simulations
(dashed line) of F1 (left) and F3 (right). ............................................................... 111
Figure 7.5.1 Comparison of fixed strength torque-driven simulation of F1 (above)
and optimal technique to produce jump height in a single straight front somersault
(below). ................................................................................................................ 121
Figure 7.5.2 Comparison of fixed strength torque-driven simulation of F2 (above)
and optimal technique to produce jump height in a 1¾ piked front somersault
(below). ................................................................................................................ 122
Figure 7.5.3 Comparison of fixed strength torque-driven simulation of F3 (above)
and optimal technique to produce jump height in a piked triffus (below). .......... 123
Figure 7.5.4 Comparison of the joint angle time histories during optimal jump
height technique (solid lines) and the fixed strength matched simulation (dashed
lines) of F1 (top), F2 (middle), and F3 (bottom). ................................................... 125
Figure 7.5.5 Comparison of activation time histories of the optimal solution for
jump height (solid line) and matched simulation (dashed line) of F1. ................. 127
Figure 7.5.6 Comparison of activation time histories of the optimal solution for
jump height (solid line) and matched simulation (dashed line) of F2. ................. 128
Figure 7.5.7 Comparison of activation time histories of the optimal solution for
jump height (solid line) and matched simulation (dashed line) of F3. ................. 129
Figure 7.5.8 Comparison of the joint torques during optimal jump height
technique (solid lines) and the fixed strength matched simulation (dashed lines) of
F1 (top), F2 (middle), and F3 (bottom). ................................................................. 131
Figure 7.5.9 Comparison of fixed strength torque-driven simulation of F3 (above)
and optimal technique to produce rotation potential with up to 1.075 m of travel
(below). ................................................................................................................ 134
Figure 7.5.10 Comparison of fixed strength torque-driven simulation of F3
(above) and optimal technique to produce rotation potential with up to 2.15 m of
travel (below). ...................................................................................................... 135
Figure 7.5.11 Comparison of the joint angle time histories of the 0.5L (left) and
1L (right) optimal techniques to and the fixed strength matching simulation of
skill F3. ................................................................................................................. 137
xiv
Figure 7.5.12 Comparison of activation time histories of the 0.5L optimal solution
for rotation potential (solid line) and matched simulation of F3 (dashed line). ... 139
Figure 7.5.13 Comparison of activation time histories of the 1L optimal solution
for rotation potential (solid line) and matched simulation of F3 (dashed line). ... 140
Figure 7.5.14 Comparison of the joint torque time histories of the optimal
techniques, 0.5L (left) and 1L (right) for the production of rotation potential (solid
line) and the fixed strength matched simulation of skill F3 (dashed line). .......... 142
xv
List of Tables
Table 3.2.1 List of marker offsets measured. ......................................................... 40
Table 3.2.2 List of recorded performances. ........................................................... 41
Table 3.2.3 Body segmental inertia parameters calculated using the inertia model
of Yeadon (1990b). ................................................................................................ 42
Table 3.2.4 Dimensions of the two-segment foot. ................................................. 43
Table 3.3.1 Mean segment lengths calculated from joint centre position data. ..... 45
Table 3.3.2 Differences in segment lengths between the inertia model and static
trials. ....................................................................................................................... 46
Table 3.3.3 Minimum and maximum joint angles achieved in the recorded
performances. ......................................................................................................... 51
Table 5.2.1 Initial conditions of seven selected skills. ........................................... 67
Table 5.2.2 Common parameter values determined by the combined matching
procedure. ............................................................................................................... 69
Table 5.2.3 Adjustments made to the initial horizontal, vertical, and angular
velocities for the individual simulations. ............................................................... 69
Table 5.2.4 Cost function component scores of angle-driven simulations. ........... 70
Table 5.3.1 Initial conditions of two skills used in the evaluation procedure. ....... 71
Table 6.3.1 Torque generator strength parameter values. ...................................... 81
Table 6.3.2 Seven parameters defining muscle activation profiles. ....................... 83
Table 6.4.1 Upper and lower bounds for muscle activation parameters. ............... 90
Table 6.4.2 Limits of the range of motion of the joints, as used for penalties. ...... 93
Table 6.4.3 Cost function component scores resulting from the strength scaling
matching protocol. ................................................................................................. 94
Table 6.4.4 Strength scaling factors. .................................................................... 103
Table 6.4.5 Cost function component scores resulting from the fixed strength
matching. .............................................................................................................. 104
Table 7.5.1 Differences between matched simulations (M) and the optimal
simulation (O) for maximum jump height in the skills F1, F2, and F3................. 120
Table 7.5.2 Differences between matched simulations (M) and the optimal
simulation (O) for maximum rotation potential for two limits of travel 0.5L and
1L. ........................................................................................................................ 133
1
Chapter 1: Introduction
Trampolining consists of alternating contact and flight phases, with somersaulting
and twisting skills performed during the flight phases. In competitive
trampolining, a sequence of ten different skills is performed in consecutive flight
phases. The skills performed comprise different combinations of somersault
rotation and twist rotation. A routine is scored based on a number of factors,
including travel (the horizontal distance between consecutive landing positions) as
well as the shape of the body during the skill.
During the flight phase it is possible for the trampolinist to control the speed of
rotation through changes in body configuration which affect the moment of inertia
of the body. Whilst airborne the trampolinist is unable to effect momentum
changes affecting the movement of his centre of mass or his angular momentum.
The skills performed in the flight phases are therefore limited by the orientation,
configuration and angular and linear momenta of the trampolinist at the moment
of takeoff and so it is during the contact phase that the correct motions must be
made by the trampolinist to allow the trampolinist to perform the next skill.
In trampolining, the duration of the contact can be separated into two discrete
phases: the depression and recoil phases, with the point of maximum bed
depression forming the transition between the two. As the trampolinist depresses
the trampoline bed there is a transfer of energy from the trampolinist to the
trampoline, through which kinetic and gravitational potential energy is stored as
elastic potential energy in the trampoline. During the recoil phase most of the
elastic potential energy in the trampoline is released and transferred back to the
trampolinist as the trampolinist is accelerated upwards.
During the recoil phase the trampolinist initiates rotation for the subsequent flight
phase using hip flexion for forward somersaults and hyperextension of the back
and hips for backward somersaults. However any flexion of the knees or hips
during the recoil phase will absorb energy resulting in lower linear momentum for
the subsequent flight phase. The compromise between height and rotation has also
2
been recognised and investigated in sports such as diving, but research on the
takeoff phase of trampolining is limited.
1.1 Previous Trampolining Research
Early literature on the topic of trampolining or ‘rebound tumbling’, as it was also
known, was based primarily on the observations and opinions of coaches with
very little scientific basis or objective data in support. These texts are based
predominantly around methods and progressions for teaching skills (Davis &
McDonald, 1980; Horne, 1968; LaDue & Norman, 1954; Walker, 1983) but
occasionally offer advice on the details of technique (Davis & McDonald, 1980;
LaDue & Norman, 1954; Walker, 1983; Phelps & Phelps, 1990).
Early research of trampolining studied the effects of impulse on momentum in
trampolining (Shvartz, 1967), the force-depression relationship of the trampoline
bed (Lephart, 1971), the movements of the centre of mass during the contact
phase (Lephart, 1972) and the kinetic properties of trampoline skills (Vaughan,
1980).
The majority of recent trampolining research has focused on the injury risks and
prevalence of injuries sustained during recreational trampolining (Black &
Amadeo, 2003; Furnival et al., 1999; Larson & Davis, 1995; Murphy, 2000;
Smith & Shields, 1998) and one research group has studied the effects of
trampolining on sleep (Buchegger & Meier-Koll, 1988; Buchegger et al., 1991),
with some research having been dedicated to further understanding the mechanics
of trampolining skills (Ollerenshaw, 2004), and some effort has been made to
understand the physical properties of the trampoline (Jaques, 2008; Kraft, 2001).
Studies of the mechanics of trampolining are limited. Lephart (1972) explained
how somersaults can be performed without travel by considering the motion of the
centre of mass throughout the entire contact phase. He found that in forward
somersaults the mass centre is travelling forwards at the time that the backwards
force is applied; this force then decelerates the velocity centre of mass to zero at
takeoff to produce rotation and eliminate the horizontal movement in flight. A
3
study by Ollerenshaw (2004) developed a method by which the contributions of
horizontal and vertical reaction forces to the production of angular momentum in
forward somersaults could be quantified. This study also developed a method for
allocating force between foot contact locations for different levels of depression of
the trampoline.
1.2 Statement of Purpose
The purpose of this study is to gain an understanding of the mechanics used to
generate linear and angular momentum during the contact phase of trampolining
and to investigate optimal technique for the production of somersault. This task
will be made possible by the use of a computer simulation model of the
trampolinist and the trampoline throughout the contact phase. The simulation
model will then be evaluated before investigating the techniques used to create the
required momenta for different skills and investigating optimal technique for the
production of both forward and backward somersaults.
1.3 Research Questions
If air resistance is negligible, during the flight phase no external forces other than
gravity act on the trampolinist and so at takeoff the trampolinist must have the
required angular momentum with maximum vertical linear momentum and little
or no horizontal linear momentum. The production of angular momentum requires
energy from the trampoline that would otherwise be transferred into linear kinetic
energy, reducing the vertical linear momentum (Miller & Munro, 1984). The
interrelated nature of angular and linear momenta means that an optimal solution
should produce the angular momentum required without compromising vertical
takeoff velocity allowing the trampolinist to reach a maximal peak height.
Q1. For specific skills with a fixed rotational requirement what is the optimal
takeoff technique to produce the required angular momentum with maximum peak
height and minimum travel?
4
The interrelationship between angular and linear momenta produced during
trampolining takeoff has been demonstrated (Miller & Munro, 1984; Sanders &
Wilson, 1988). In order to maximise rotation potential (i.e. the product of angular
momentum and flight time) the trampolinist must rotate quickly whilst staying
airborne for a suitably long period of time. The greater the degree of
flexion/hyperextension of the trampolinist during the contact phase and at takeoff,
causes the reaction force to be more off-centre and more energy is used to create
angular momentum rather than linear momentum resulting in less time in the air to
complete rotation (Cheng & Hubbard, 2004, 2005; Sanders & Wilson, 1988,
1992). In order to complete the maximum amount of somersault rotation a
technique that balances the production of linear and angular momenta must be
found.
Q2. What is the optimal takeoff strategy to produce maximal somersault rotation
potential in forward somersaults?
1.4 Chapter Organisation
Chapter 2 critically reviews the literature on the biomechanics of trampolining.
The reviewed literature includes theoretical and experimental research studies, as
well as coaching publications, identifying limitations of previous work and
highlighting gaps in the research area.
Chapter 3 describes the methods used to collect kinematic performance data and
anthropometric measurements from a trampolinist. The procedures employed to
process and analyse the kinematic and anthropometric data are detailed, and
graphic sequences of the trampoline movements are presented.
Chapter 4 describes the structure and function of the computer simulation model
and the interactions with the trampoline suspension system. The kinetics of the
trampolinist-suspension system interactions are discussed, and a method for
determining the centre of pressure on the foot is outlined.
5
Chapter 5 details the application of the angle-driven computer simulation model
of the trampolinist. The procedure for employed for the determination of the
visco-elastic parameters is detailed and the angle-driven model is evaluated.
Chapter 6 details the structure and function of the torque-driven computer
simulation model of the trampolinist. The torque-driven model is applied and the
protocols used to scale the strength of the simulation model and to evaluate the
model with fixed strength are detailed.
Chapter 7 describes the application of the torque-driven model to optimise
technique and answer the research questions. The results of the optimisation of
technique are reported and analysed.
Chapter 8 provides a summary of the thesis. The research questions are answered
and the results obtained are summarised. The methods used in the study are
discussed, and potential future applications of the simulation model are outlined.
6
Chapter 2: Review of Literature
2.1 Chapter Overview
In this chapter literature on the topics of trampolining and simulation modelling
are reviewed. Literature concerning other areas specific to this study are also
discussed and gaps in the literature are highlighted.
2.2 Overview of Trampolining Research
Early literature on the topic of trampolining was based primarily on the
observations and opinions of coaches with very little scientific basis or objective
data in support. Early texts are predominantly coaching literature focusing on
methods and progressions for teaching skills (Davis & McDonald, 1980; Horne,
1968; LaDue & Norman, 1954; Walker, 1983) but occasionally offer advice on
the details of technique (Davis & McDonald, 1980; LaDue & Norman, 1954;
Walker, 1983; Phelps & Phelps, 1990).
Early research of trampolining examined the interactions between the trampolinist
and trampoline; the effects of impulse on momentum in trampolining (Shvartz,
1967), and the force-depression relationship of the trampoline bed (Lephart,
1971). Research later progressed to studying the influence the trampolinist had on
this interaction and how the trampolinist utilised the properties of the trampoline.
Lephart (1972) examined the movements of the centre of mass during the contact
phase and Vaughan (1980) investigated the kinetic properties of trampoline skills.
The majority of recent trampolining research has focused on the injury risks and
prevalence of injuries sustained during recreational trampolining (Black &
Amadeo, 2003; Furnival et al., 1999; Larson & Davis, 1995; Murphy, 2000;
Smith & Shields, 1998) and one research group has studied the effects of
trampolining on sleep (Buchegger & Meier-Koll, 1988; Buchegger et al., 1991),
with some research having been dedicated to understanding the mechanics of
trampolining skills (Ollerenshaw, 2004), and some effort has been made to
7
understand the physical properties of the trampoline further (Jaques, 2008; Kraft,
2001).
Investigations into the mechanics of trampolining are limited however. Studies
such as those by Lephart (1972) and Ollerenshaw (2004) provide an insight into
the mechanics of trampolining although there is still large scope for this to be
expanded upon.
2.2.1 Takeoff
In trampolining the contact phase consists of five distinct elements: the moment of
touchdown, the depression phase, the moment of maximal depression, the recoil
phase and the moment of takeoff. Throughout the contact phase the trampolinist
adjusts his motion so that at takeoff the trampolinist possesses sufficient linear
and angular momentum to complete the next skill successfully. These two factors
are required in order to provide the trampolinist sufficient flight time and angular
momentum to complete the skill before touchdown for the subsequent contact
phase, during which this process is repeated. It is necessary for a trampolinist to
address his momenta prior to takeoff as during flight the only force acting on the
trampolinist, gravity, is constant and so the motion of his centre of mass cannot be
altered, but changes in his body configuration can be used to control the speed of
rotation about the centre of mass. Therefore by the instant of takeoff the
trampolinist must possess sufficient vertical velocity and sufficient angular
momentum in order to complete the skill.
2.2.2 Height
In trampolining the peak height during flight does not currently affect directly the
marks given to a routine although the maintenance of height from one skill to the
next does receive consideration from the judges (FIG, 2013). Height is also
important in trampolining because it provides time for skills to be executed and
more than adequate height and time can give the illusion of ease that is associated
with excellent performance (Miller & Munro, 1985; Sanders & Wilson, 1988).
Biomechanical studies of trampolining, diving and vertical jumping have
identified some of the characteristics that can affect jump height. Both theoretical
8
(Cheng & Hubbard, 2004) and experimental (Shvartz, 1967) studies have
associated maximal depression of an elastic surface with maximal jump height in
both trampolining and diving.
Jump height is ultimately dependent on the vertical position and velocity of the
mass centre at takeoff, although takeoff velocity has been found to be the primary
factor in improving jump height (Feltner et al., 1999; Sanders & Wilson, 1992).
Many studies have associated an increase in takeoff velocity with an increase in
touchdown velocity (Miller, 1984; Miller & Munro, 1984, 1985). Sanders &
Wilson (1988) found that increased vertical velocity at touchdown in springboard
diving allowed more energy to be stored in the springboard and then transferred
back to the diver to increase takeoff velocity. In trampolining, however, the
depression of the trampoline is limited by the height of the frame.
Studies investigating the movement strategies used to obtain maximal height have
also been conducted in various contexts; these include squat jumps and
countermovement jumps (Pandy & Zajac, 1991) and drop jumps from both rigid
and compliant surfaces (Cheng & Hubbard, 2004, 2005; Miller & Munro, 1984,
1985; Sanders & Allen, 1993; Sanders & Wilson, 1988, 1992; Vaughan, 1980).
Miller & Munro (1984) identified the two objectives of movement strategies in
jumping from compliant surfaces as maximising the upward acceleration of the
centre of mass relative to the surface during the depression phase and minimising
the negative acceleration of the centre of mass relative to the surface during the
last part of the recoil phase. These objectives are associated with an increased
reaction force (Sanders & Wilson, 1992; Vaughan, 1980) and impulse (Shvartz,
1967) caused by maximal extension of the lower body during the depression
phase (Cheng & Hubbard, 2004, 2005; Miller & Munro, 1984; Vaughan, 1980),
used in conjunction with a well-timed arm swing (Miller & Munro, 1984), and
maintenance of this extension throughout the recoil phase (Cheng & Hubbard,
2005).
Maximal extension of the lower body is performed from an optimally flexed
position at touchdown to allow maximum range over which extension can take
place (Sanders & Wilson, 1988), with hip extension beginning before touchdown
(Sanders & Allen, 1993); however the amount of hip extension prior to
9
touchdown is limited by strength so that further flexion and energy absorption is
not caused by the impact of landing. Then during the depression phase the body is
not maximally extended in a proximal-to-distal sequence (Bobbert & Ingen
Schenau, 1988) but with initial knee extension followed by hip and ankle (Cheng
& Hubbard, 2005; Sanders & Allen, 1993; Selbie & Caldwell, 1996). The
sequence does not follow the proximal-to-distal order because the body must re-
orientate itself prior to propulsion (Selbie & Caldwell, 1996).
During the depression phase the trampolinist must also use the arms if the
trampolinist is to apply maximal force to the bed during the contact phase (Cheng
& Hubbard, 2008; Dapena, 1993; Hara et al., 2006, 2008; Lees et al., 2004; Payne
et al., 1968). Using an arm swing has also been found to increase the height of the
centre of mass at takeoff in jumping (Lees et al., 2004) and transfers momentum
to the rest of the body when the trampolinist starts to decelerate (Harman et al.,
1990). In springboard diving, Miller and Munro (1984) suggested that the arm
swing should commence before touchdown and should be beginning to positively
accelerate at this moment, the arm swing should then continue through until
takeoff or for as long as possible. However, the findings of Cheng & Hubbard
(2008) suggest that another strategy allowing greater work to be done at the hips
may be optimal.
During the recoil phase the trampolinist is accelerated upwards by the trampoline
bed until takeoff. To maximise takeoff velocity and the height reached during a
jump, the trampolinist must be able to utilise most of the energy stored in the
trampoline and avoid leaving energy in the trampoline rather than transferring it
into vertical velocity. An extended, straight posture, as well as increased stiffness
of the knee and hip, has been associated with improved jump height (Cheng &
Hubbard, 2004, 2005; Sanders & Wilson, 1988, 1992).
During the contact phase there is a continual transfer of energy between the
trampolinist and trampoline suspension system. The trampolinist performs work
on the trampoline during the depression phase, using a strong leg push and well-
timed arm swing. The trampolinist then receives this work back from the
trampoline during the recoil phase by continuing to extend in order to prevent any
energy being absorbed or left behind.
10
2.2.3 Rotation
Rotation is caused by an external force whose line of action does not to pass
through the body’s centre of mass. The size of the torque and amount of angular
momentum given to the body are dependent on the product of the magnitude of
the force and the perpendicular distance between the line of action of the force
and the axis of rotation (Stroup & Bushnell, 1969).
In trampolining the reaction force is controlled by moderating the force the
trampolinist applies to the trampoline and the distance is controlled by altering the
position of the centre of mass through changes in body configuration and
orientation during the depression phase. The generation of angular momentum has
been found to primarily take place during the recoil phase of contact (Mathiyakom
& McNitt-Gray, 2007; Miller & Munro, 1985) and it is controlled by movements
of the centre of mass (Ollerenshaw, 2004). Angular momentum can be transferred,
during the recoil phase, from remote body segments to adjacent, more proximal
body segments (Hamill et al., 1986; Cheng & Hubbard, 2008).
Body configuration can be changed to create a torque by bending at the hips
moving the centre of mass away from the line of action of the force (Aaron, 1970;
Blajer & Czaplicki, 2001; Mathiyakom & McNitt-Gray, 2007). Leaning to change
body orientation, by relaxing the plantar flexors for forward rotations, can produce
a torque, moving the centre of mass from above the base of support away from the
line of action of the reaction force (Page, 1974).
Early coaching literature on the production of angular momentum (specifically of
forward somersaults) was of the consensus that at takeoff the arms should be
thrown forward from their initial overhead position and the hips thrust backwards
by piking at the hips (Musker et al., 1968; Loken & Willoughby, 1967; Keeney,
1961). These motions cause the feet to push forward on the trampoline creating a
horizontal reaction force backwards at the feet, producing a torque about the mass
centre. These principles can also be used to explain the production of a forward
reaction force in backward somersaults where the body is hyper-extended and the
arms rotated backwards overhead (Cheng & Hubbard, 2008). One problem that
arose from early explanations of the production of angular momentum in
trampolining was the explanation of lack of gain. Whilst some authors, who were
11
advocates of leaning to produce angular momentum (Bunn, 1955), were confused
about the position and movements of the centre of mass, others did not consider
the linear effects of a non-vertical force (Griswold, 1948; Dyson, 1964).
Lephart (1972) explained how somersaults can be performed without travel by
considering the motion of the centre of mass throughout the entire contact phase.
He found that in forward somersaults the mass centre is travelling forwards at the
time that the backwards force is applied. The horizontal reaction force then
decelerates the velocity centre of mass to zero at takeoff to produce rotation and
eliminate the horizontal movement in flight. The study also concluded that
throughout the contact phase the centre of mass was always above the base of
support, this is in agreement with Frohlich (1979) and limits the contributions of
lean to the production of angular momentum.
More recent studies have shown that more complex skills require larger angular
momentum at takeoff (Hamill et al., 1986; Miller & Sprigings, 2001;
Ollerenshaw, 2004; Sanders & Wilson, 1987). Ollerenshaw (2004) investigated
the contributions of vertical and horizontal forces to the production of angular
momentum in trampoline takeoffs using an experimental approach. Ollerenshaw
(2004) concluded that the horizontal force, in agreement with the findings of
Lephart (1972), controlled travel and also developed torque through movements
of the centre of mass relative to the vertical force.
2.2.4 Inter-relationship of Height and Rotation
It has been widely recognised that both vertical velocity and angular momentum
at takeoff are critical factors for the performance of somersaults (Brüggemann,
1983, 1987; Hwang et al., 1990; King & Yeadon, 2004). In order for a
trampolinist to rotate maximally, the trampolinist must possess a large amount of
angular momentum and a large vertical velocity at takeoff, so that the trampolinist
rotates quickly and has a long flight time in which to rotate (King & Yeadon,
2004). Both the generation of angular momentum and the production of jump
height are dependent on the vertical force experienced by a trampolinist
(Ollerenshaw, 2004; Vaughan, 1980). To reach maximal height a trampolinist
must maintain a straight body position (Cheng & Hubbard, 2004) and to create
12
rotation a trampolinist must assume a piked position (Lephart, 1972). Thus in
order to perform an optimal takeoff the trampolinist must compromise between
these two factors. It is important, for the execution of somersaults, that the
technique used does not produce flight time at the expense of angular momentum,
and vice versa; hence optimal technique will achieve a balance between the two.
Both experimental studies of diving (Miller & Munro, 1984; Sanders & Wilson,
1988) and theoretical calculations (Stroup & Bushnell, 1969) have observed this
inter-relationship. These studies found that as the somersault requirement
increased there was a reduction in jump height.
The production of angular momentum requires changes in body shape but such
movements detract from jump height, and therefore to produce optimal
performance it is sensible that the movements must be performed with the correct
sequencing, timing and magnitude (Kong, 2005). It has been suggested that for the
similar motion of diving takeoffs, the optimal technique may vary depending on
the individual qualities and preferences of a diver (Xu & Zhang, 1996; Xu, 2000).
Kong (2005) modelled diving takeoffs with the aim of finding an optimal
technique. Little is known however about optimal technique, in terms of height
and rotation, for trampolining takeoffs.
2.2.5 Summary
It has long been established that the production of angular and linear momenta
during the takeoff phase is of primary importance for any airborne activity. As the
rotation requirement of the activity increases, the trampolinist must alter his
movement patterns accordingly, although the extent to which these patterns need
to be altered is unknown. The compromise between angular and linear momenta is
well established, but no investigation as to the optimal balance of these factors in
trampolining has been made.
2.3 The Trampoline
Competitive trampolines consist of a bed of webbed nylon suspended within a
steel frame by approximately 120 steel springs. The frame of a trampoline stands
13
approximately 5 m x 3 m x 1 m with the bed measuring approximately 4 m x 2 m.
Trampoline beds can have various sizes of webbing, ranging between 25 mm and
4 mm wide, with the width and spacing of the webbing affecting the stiffness of
the bed. The springs have a natural length of approximately 35 cm, and suspend
the bed in tension by hooking on to the frame and bed along both the length and
breadth of the trampoline.
2.3.1 Modelling the Trampoline
Few attempts have previously been made to model the trampoline (Blajer &
Czaplicki, 2001; Jaques, 2008; Kraft, 2001; Lephart, 1971); more effort has been
made to model other compliant surfaces, such as springboards (Kong, 2005; Kooi
& Kuipers, 1994; Sprigings et al., 1989) and crash mats (Mills, 2005).
Lephart (1971) made an early attempt to understand the force-displacement
relationship of the trampoline by taking static measurements of the upward pull of
the trampoline bed when depressed using scales. By measuring the force at one
inch intervals up to a depression of twenty inches, a non-linear vertical force-
displacement relationship was identified. In other early analyses of trampolining,
Riehle (1979) and Vaughan (1980) modelled the vertical motion of a trampolinist
as simple harmonic motion, and Vaughan (1980) included damping to account for
discrepancies in the accelerations. However, Kraft (2001), in agreement with
Lephart (1971), believed that the trampoline could not be accurately represented
by an ideal Hookian spring due to the construction of the trampoline, even if the
trampoline springs were themselves linear springs, due to the changing angle of
the springs throughout the contact phase.
Kraft (2001) developed a theoretical equation for the vertical force, Fv, exerted by
a trampoline at a given depression based on the physical geometry of the
trampoline as a cross-section across the trampolines width. The relationship he
derived is shown in Equation 2.2.1.
14
(2.2.1) 𝐹𝑣 = −𝐷.𝑠.[𝑙0 + √(𝑠2 + 𝑏2) − 𝑏]
√(𝑠2 + 𝑏2)
Where: s = vertical depression of the bed
D = stiffness of the springs
l0 = stretch of the springs at equilibrium
b = distance from the frame to the nearest point of action
This model combined the 20 springs on either side of the trampoline to represent
them as a single spring. The model was then used to investigate vertical trajectory
and contact time in trampolining.
Blajer and Czaplicki (2001) investigated the vertical, horizontal and rotational
force-displacement relationship of a trampoline in order to understand the motion
of a gymnast. The study found the vertical force to be non-linear with respect to
displacement and a linear horizontal force-displacement relationship.
Kennett et al. (2001) used finite element modelling to simulate a recreational
trampoline as a system of 7948 isolinear 4-noded membrane elements suspended
by 88 springs. In experimentation with various types of spring model it was
discovered that the pretension of the trampoline is a vital element of the response
of the trampoline bed. However the model was unsuccessful as the software used
was unable to add the required level of damping, resulting in high frequency
vibration of the trampoline bed, much like a drum skin.
All the previous studies have neglected the inertial characteristics of the
trampoline itself, and their effect on the force-displacement relationship. Jaques
(2008) modelled the trampoline as a system of linear springs and point masses
based on experimental measurements and found that the vertical and horizontal
force-displacement relationships were non-linear and linear respectively with the
horizontal force also dependent upon the vertical displacement. The trampoline
model was a system of thirty-eight undamped linear springs and 15 point masses
in order to represent the stiffness and inertial properties of the bed and springs of
the trampoline. Such complexity allowed this model to simulate off centre
impacts, although with limited resolution. The springs that represented the elastic
properties of the bed and springs were given different stiffnesses to characterise
15
(2.2.2)
their different properties, and multiple trampoline springs were represented in this
simplified model with a single spring of increased stiffness. This model was
evaluated using forces measured by force transducers during dynamic tests, and
was found to closely represent the vertical motion of the bed. It was less accurate
however in modelling horizontal forces.
Other compliant surfaces have been modelled by masses and springs in multiple
arrangements. Sprigings et al. (1989) modelled a springboard as a single mass and
undamped linear spring in order to represent its vertical motion, whilst Kooi and
Kuipers (1994) developed a model that consisted of a series of torsional springs
connected by solid bars; this modelled the motion of the springboard in two
dimensions. Kong (2005) modelled the springboard as a rod with vertical,
horizontal and rotational degrees of freedom. Mills (2005) modelled gymnastic
landing mats as three damped linear spring and masses in series representing the
different component layers.
2.3.2 Measuring Trampoline Parameters
The equation derived in Kraft (2001) required three parameters to be determined:
the stiffness of the springs, the stretch of the springs with the bed at equilibrium
and the shortest distance from the frame to the point of force application. These
parameters were determined experimentally with the use of simple static length
and displacement measurements. The stiffness, k, of the springs can be determined
by the application of Hooke’s Law:
𝐹 = 𝑘. 𝑦
Where: F = applied load
y = resulting extension
The lengths of eight randomly selected springs were measured when in situ with
the trampoline at rest before being removed from the trampoline and measured
with loads of 0, 5, 10, 15 and 20 kg. From these measurements the mean stiffness
of each spring was determined to be 703 Nm-1
, this relationship was extrapolated,
correcting for weight, to find that the natural length of the springs and their
extension when the trampoline is at rest. The trampoline was also tested as a
16
whole by loading centrally positioned trays of 20 cm x 20 cm with 15 loads up to
362 kg and 40 cm x 50 cm with five loads between 153 kg and 533 kg. The
vertical stiffness of the trampoline for different contact areas was then determined
by minimising the sum of error squares for the measurements from the two trays;
this spring constant is the stiffness of the trampoline bed and is different to the
spring constant, k.
Simple laboratory tests were used by Kennett et al. (2001) to determine the
parameters of their model. They conducted drop tests over the area of the
trampoline bed and took measurements using a string potentiometer to measure
time-displacement properties. Although the method used by Blajer and Czaplicki
(2001) to determine input parameters is unclear, it seems that known loads were
applied and the resulting displacements were measured as well as damping
coefficients being observed from vibrations of the trampoline bed. The authors
acknowledged that better measurements of bed deflection during performance
were required to improve the accuracy of the model.
The input parameters for the model of Jaques (2008) were determined through
static measurement of the stiffness of the springs and length and mass
measurements of the bed and springs in situ and in their natural state. The
dimensions of the bed and springs were measured whilst under tension before
being removed from the trampoline to be weighed and to measure the dimensions
of the bed and length of the springs under no tension. The springs were then
loaded with known masses up to 35 kg, to give a range of extension representative
of those experienced by the springs during trampolining. The extension of the
springs under each load was measured and the stiffness of the springs was
calculated using Hooke’s Law. In the construction of the model the mass of the
bed and springs was distributed between the point masses depending on the mass
of the components represented by the elements connected to that specific mass.
The stiffnesses of the springs representing the bed were calculated with extensions
taken from the difference between the length measurements taken under tension
and in the natural state, assuming they were in equal tension with the springs
connected at either end.
17
2.3.3 Limitations of Previous Trampoline Models
Previous models of trampolines have either given estimates of forces relative to
displacements (Lephart, 1971), only modelled the vertical component of force
(Kraft, 2001; Lephart, 1971), not been properly evaluated (Blajer & Czaplicki,
2001; Kennett et al., 2001; Kraft, 2001; Lephart, 1971), or have neglected the
effect that the mass of the trampoline suspension system, and the force required to
accelerate this mass, affects the reaction force experienced by the trampolinist.
2.4 Computer Simulation Models
2.4.1 Overview
Computer simulation of human movement involves the description of motion
through the development and application of mathematical equations and can help
us to understand the mechanics of a sporting movement. The processes required to
develop a computer simulation model include the definition of the problem, the
derivation of the governing mathematical equations, the writing of the computer
program, the determination of input values, validation of the model, and the
completion of simulation experiments (Vaughan, 1984). The use of simulation
models is very useful in experiments where variables can be changed and
controlled in a way that would be impossible in experimental studies, allowing
new techniques to be investigated without the risk of injuring an elite athlete.
However useful computer simulation models may be to investigate physical
phenomena, they require expertise in both the field of mathematics and computer
programming to develop and implement (Vaughan, 1984) and the result is still
only a group of mathematical equations that may or may not be representative of
the problem at hand or have sufficient accuracy to answer the problem posed
(Panjabi, 1979; Sargent, 2005). Specialised computer programs for the
construction and development of simulation models are commercially available to
help derive the mathematical equations (e.g. AUTOLEV, DADS, MADYMO),
and have broadened the scope of simulation modelling. However, knowledge of
the movement, modelling and an understanding of the processes is essential to
apply the software correctly and interpret its output. If mathematical theory and
18
the modelling process are applied correctly the resulting model should be
representative of the problem but this cannot and must not be assumed. All
simulation models should be verified and validated before any findings from the
model are trusted (Sargent, 2005). Yeadon et al. (1990) performed such an
evaluation through the comparison of the performance outcome from their
simulation model to experimental data obtained from a performance of the
trampoline skill upon which the simulation was matched.
2.4.2 Trampolining Simulation Models
Simulation models have been used in trampolining in order to learn about
numerous aspects of the sport, including the determination of joint torques (Blajer
& Czaplicki, 2001, 2003) and internal forces during contact (Blajer & Czaplicki,
2003, 2005), to understand the controlling movements in somersaults (Flynn &
Simms, 2003), as well as more technical usage in the development of
trampolining robots (Takashima et al., 1998) and image recognition systems
(Kikuchi & Nakazawa, 2004). However progress has been slowed by the
complexity of modelling the trampoline itself and the only study to utilise a
specialised trampoline model is that of Flynn and Simms (2003). Simulation work
of similar sports, such as gymnastics and diving, that do not require such a
complex prerequisite as well as aerial movement in general, are much more
advanced (Cheng & Hubbard, 2008; King & Yeadon, 2004; Yeadon, 1990a, b, c;
Yeadon et al., 1990).
Previous studies on trampolining by Yeadon (1990) have been in two dimensions
with the exception Kikuchi & Nakazawa (2004) who used a three-dimensional
simulation model consisting of sixteen segments with thirty-one degrees of
freedom (DOF). This model was only used to track the motion of a trampolinist
during the flight phase and so did not require feet, however the spine was
modelled as two segments along with a separate segment for the pelvis.
Takashima et al. (1998) developed a planar three-segment, 5 DOF model actuated
by torque generators to be used in conjunction with a robot in order to try and
replicate the repeated bouncing of a trampolinist. The segments represented the
lower leg, thigh and trunk of a trampolinist, and the torque generators were simple
19
to represent the motors of the robot rather than the complex torque development
profile of human muscle. The aim of the simulation was to perform jumps without
rotating so that it did not fall down, jump off the trampoline or lose height. This
simple model was able to achieve this for over 100 consecutive jumps. The robot
however could only perform ten jumps, showing that there were differences
between the model and the robot.
Blajer and Czaplicki (2001, 2003, 2005) used a planar, seven-segment rigid link
system driven by six torque generators in order to investigate the joint torques and
internal joint forces that act during the contact phase. The model assumed that
bilateral limb movements were synchronous and used a single segment to
represent the foot. In this study the joint torques were calculated by inverse
dynamics, or similar method, for use in the forward dynamics simulation. This
model, although being simple, performed well and demonstrated that joint torques
determined by inverse dynamics can be used as input for a forward dynamics
model, although this has limitations in the modelling of hypothetical movements.
A simulation model was constructed by Flynn and Simms (2003) in order to study
the rotation of back somersaults and was used in conjunction with a Kraft (2001)
trampoline model. The model was angle-driven and consisted of eight segments,
including single segment feet but no arms, the mass of the arms being included in
the torso. The model was driven by joint angle time histories from digitised video
of performances of ¾ and 1¼ back somersaults having been given the same initial
conditions to investigate the effects of changes in the body configurations.
Previous simulation models of trampolining have been relatively simple and have
not attempted any kind of optimisation of performance, possibly due to the lack of
an accurate model representing the important characteristics of the trampoline.
The aims of trampolining are quite clear, to produce linear and angular
momentum whilst controlling the horizontal motion of the centre of mass. The
construction of a satisfactory model of the forces exerted by a trampoline on a
trampolinist could be combined with a trampolinist in order to answer more
complex questions concerning the movements required during the contact phase
in order to produce optimal performance.
20
2.4.3 Non-trampolining Takeoff Simulation Models
A large volume of research has been conducted using computer simulation of
takeoffs in various sporting contexts, ranging from standing vertical jumps [e.g.
basketball] (Anderson & Pandy, 1999; Dapena, 1999; Feltner et al., 1999;
Haguenauer et al., 2006; Jacobs et al., 1996; Pandy et al., 1990; Selbie &
Caldwell, 1996; Spägele et al., 1999; van Soest et al., 1993; Virmavirta, et al.,
2007), to running vertical and horizontal jumps [e.g. high jump and long jump]
(Alexander, 1990; Dapena & Chung, 1988; Hatze, 1981a; Mesnard et al., 2007),
jumping from compliant surfaces [e.g. springboard diving] (Boda, 1992; Cheng &
Hubbard, 2004, 2005, 2008; Kong, 2005; Liu & Wu, 1989; Sprigings et al., 1986;
Sprigings & Miller, 2002; Sprigings & Watson, 1985) and somersaulting jumps
[e.g. gymnastics] (Hamill et al., 1986; King & Yeadon, 2004; Yeadon & King,
2002). Whilst being applied in very different ways many of these models are
similar in their construction often with slight changes to apply a certain goal of
performance. These models of similar tasks can provide an important insight into
modelling trampolining takeoffs.
Some studies have used extremely complicated simulation models in an attempt to
reproduce the kinetic, kinematic and muscular patterns of vertical jumps
(Anderson & Pandy, 1999; Hatze, 1981a; Spägele et al., 1999). Hatze (1981a)
used a 17-segment muscle-driven model that had 42 DOF and 46 muscle groups
to simulate long jump takeoffs. This is still one of the most comprehensive models
used to date although the amount of input data required to run the model led it to
be a time consuming process. Anderson and Pandy (1999) developed a three-
dimensional, 10-segment model of vertical jumping with 23 DOF powered by 54
muscle groups that was able to accurately reproduce all elements of a vertical
jump (Pandy & Anderson, 2000).
Yeadon and King (2002) developed a subject-specific, five-segment, torque-
driven model of tumbling which was personalised through the determination of
subject-specific strength parameters (King & Yeadon, 2002). The same model
was later used in other studies to investigate the robustness of the model to
perturbations in layout somersaults (King & Yeadon, 2003) and to optimise
somersault performance in tumbling (King & Yeadon, 2004).
21
An eight-segment model with eight DOF, with wobbling masses and actuated by
torque generators was used by Kong (2005) to optimise the performance of a
diver. The model had a two-segment foot and the foot/springboard interface was
modelled using three pairs of perpendicular damped linear springs. The model
was personalised to the diver through inertia, torque and visco-elastic parameters,
and was used in conjunction with a springboard model in an optimisation process
which first matched simulations to performances and then found a strategy to
produce an optimal amount of rotation during flight. Values for the input
parameters describing the elastic properties of the foot contact for the torque-
driven model were extremely difficult to determine experimentally and so these
were obtained through a matching procedure using an angle-driven model. This
model was able to increase angular momentum by 28% and increase maximum
height by two centimetres.
2.4.4 Wobbling Mass Models
The majority of biomechanical models of the human body are composed of rigid
segments; however the human body is not rigid and is composed of soft flesh
surrounding a rigid bone structure. The soft flesh comprises muscle and organs
held in place by connective tissues with elastic properties, whilst the muscles
themselves alter the elastic properties when activated. Cavagna (1970) conducted
an early study to investigate the elastic properties of the body during a landing on
the balls of the feet with the calf contracting, following a small vertical jump. The
stiffness of the elastic structures of the body was measured by observing the
oscillatory motion of the body and was found to increase with the load on the
body in a similar relationship to that of the series elastic component of muscle.
Nigg and Liu (1999) used a wobbling mass model to simulate impacts during
running and to investigate peak GRF. The model used two pairs of rigid and
wobbling masses to represent the supporting leg and the rest of the body linked by
a combination of springs and spring-damper units to represent the series elastic
components and contractile elements, respectively, of muscle tendon units. The
model was able to closely match experimental force measurements, and was used
to conclude that the elastic properties of the connections between the soft and
rigid elements of the model had a strong influence on the peak impact forces.
22
Similarly, Gruber et al. (1998) conducted a comparative investigation of rigid
segment and wobbling masses, looking at the ground reaction force (GRF) and
joint torques produced by the two models. The models consisted of three segments
with identical inertia parameters, however the wobbling mass model separated the
mass of each segment into fixed and wobbling parts that were coupled with quasi-
elastic and strongly damped connections. The study showed that rigid body
models are inadequate to represent the GRF and internal torques during impact
and that the use of a forward dynamics rigid body model could lead to systematic
over-estimations of GRF and joint torques.
The use of wobbling mass models has continued in modelling impacts of the
lower leg, investigating the effects of a heel pad, the elastic properties of the rigid
mass-wobbling mass connection and the segmental bone-soft tissue ratios on
GRF, energy dissipation, joint torques and joint reaction forces. Pain and Challis
(2001) used a model combining wobbling masses with a heel pad and a
deformable knee to investigate the effect of all soft tissue in lower leg impacts.
The knee and heel pad were modelled by spring dampers in series with the lower
leg represented by a wobbling mass coupled to the rigid body by non-linear
translational spring damper actuators. This model was then impacted by a
pendulum to study the energy dissipated during impacts. It was concluded that
both the heel pad and the wobbling mass play vital roles in the dissipation of
energy during impacts.
The same research group used a three-segment model with wobbling masses and a
heel pad to investigate the sensitivity of the simulated GRF and thigh angle to
changes in the model’s parameters (Pain & Challis, 2004). The parameters were
changed by ±20% and whilst the segmental bone-soft tissue mass ratios and joint
stiffnesses were found to have large effects, changes in the stiffness of the
connection between the bone and soft tissue had little effect. Subsequently the
same model was used with subject-specific input parameters to investigate GRF,
joint torques and joint reaction forces during impacts (Pain & Challis, 2006).
Subject-specific inertia parameters were determined, the mass ratios of the rigid
and wobbling masses were based on cadaver data of Clarys & Marfell-Jones
(1986), a heel pad model was included and the spring-damper connection
23
parameters and initial kinematics were experimentally determined. This model
was able to simulate the GRF to within 5% over the first 40 ms and within 12%
over the first 100 ms, as well as matching joint angles to within 3° after the first
40 ms.
The majority of research using wobbling mass models has been in simulating
impacts with rigid surfaces, however recently some investigation into impacts
with compliant surfaces have taken place. Yeadon et al. (2006b) determined a
single set of subject-specific input parameters for an eight-segment, wobbling
mass model of springboard diving to be used for multiple performances by
combining the visco-elastic parameters determined for individual performances in
order to generalise them for multiple performances. The results of the model were
tested for sensitivity to changes in the wobbling mass parameters, and a change of
less than 1.5% was found for increases of 500 times in all the wobbling mass
parameters. This result suggests that simulating jumping from compliant surfaces
does not require the inclusion of wobbling masses. Wilson et al. (2006) also used
a combined matching approach for the determination of visco-elastic parameters
for use in a wobbling mass simulation of running jumps. Mills et al. (2008) has
also used a simulation model including wobbling masses to study the influence of
model complexity on estimates of internal loading in gymnastics landings onto a
compliant mat.
2.4.5 Summary
Simulation modelling is an extremely useful tool when answering speculative
questions, and has become increasingly popular in sports biomechanics as
advancements in technology have been made. The level of complexity that is
required in the simulation model is dependent on the level of accuracy required to
answer the questions of the researcher. Assumptions made in the creation of the
model can be validated when the model is evaluated by attempting to match an
actual performance. After validation the model can be applied to situations for
which it has been validated to investigate and optimise techniques.
24
2.5 Simulation Model Input
Simulation models require parameters as an input to provide information
concerning the initial conditions of a simulation, the characteristics of the model’s
components, the capabilities of the model and how the different components of
the model interact. Model parameters, such as strength and inertia parameters can
be determined experimentally, through the application of another model or
through optimisation when the information required cannot be measured directly.
2.5.1 Strength Parameters
In a torque-driven or muscle-driven simulation model the strength parameters of
the model greatly affect the resulting simulations of human movement; therefore it
is important that the strength parameters are representative of the subject and the
capabilities of the muscles about the joints. In order to achieve this, muscle
models have been developed and are personalised to subjects using methods
described in this section.
2.5.1.1 Muscle Modelling
Human muscle is a complex structure, comprising many elements which have
their own complexities, and is activated through electrical signals in the nervous
system. Muscle reacts to the same activation differently under different conditions
involving length, velocity of shortening and previous activation history. For over
70 years researchers have been endeavouring to understand the properties and
relationships associated with muscular contractions and studies have investigated
the behaviour of muscle at microscopic and whole-muscle levels.
An initial attempt made to investigate the behaviour of muscle was made by Hill
(1938), through investigations of the thermodynamics of muscle action under
controlled experimental conditions. A frog muscle was tetanically activated under
isolated conditions and observations of a relationship between the force produced
by the muscle and the velocity at which it was shortening during the contraction
were made. The results of these experiments were used as the basis for a model
with a structure that represented the muscle along with the connective tissues as a
contractile element (CE), a series elastic element (SE) and a parallel elastic
25
(2.4.1)
element (PE). The CE represents the muscle fibres, the SE represents the tendons
and other elastic connective tissues that are connected in series with the muscle
fibres, and the PE represents the passive elastic properties of the muscle fibres
along with the elastic connective tissues that encase the muscle fibres.
The experimental results of Hill (1938) were used to derive a hyperbolic function
describing the force-velocity relationship of the muscle during concentric
contraction, which took the form:
(𝐹 + 𝑎) (𝑣 + 𝑏) = (𝐹𝑚𝑎𝑥 + 𝑎) 𝑏
Where: F = tensile force produced by the muscle
v = shortening velocity of the muscle
Fmax = maximum tensile force produced by the muscle
a, b = constants
Hill (1938) was able to express a complex function of human muscle successfully
using a simple equation. This has enabled the widespread use of Hill-type muscle
models and allows many muscle models to be used in one simulation, leading to
the development of complicated muscle driven simulation models (Hatze, 1981a).
The classical models have been tailored for use in many simulation models with a
number of different modifications and structures being used (Bobbert et al., 1986;
Hatze, 1981a) and also with the addition of activation profiles that have attempted
to simulate EMG profiles (Pandy et al., 1990; Rácz et al., 2002). Hawkins and
Smeulders (1998, 1999) modified a Hill-type model to translate the force-velocity
relationship of muscle into a torque-velocity relationship for multiple muscles
acting about a joint whilst also incorporating an activation profile. Recent
investigations into the accuracy of a Hill-type model to different types of muscle
have found that modifications have to be made to the model in order to account
for the proportions of fast and slow twitch muscle fibres within the muscle
(Raikova & Alodjov, 2005; Stojanovic et al., 2007). Scovil and Ronsky (2006)
studied the sensitivity of a Hill-based model to perturbations in the model
parameters, the authors discovered that whilst the muscle model is very sensitive
26
to changes in a large number of parameters, simulations of running and walking
were not as sensitive and were affected by changes in fewer parameters.
Further investigation into the contraction characteristics of muscle has led to the
determination of the force-velocity relationship in eccentric contractions as well
as the discovery of a force-length relationship. Throughout a complete range of
velocities the force-velocity relationship of muscle has been found to be double-
hyperbolic in nature (Edman, 1988) with greater forces generated in maximal
eccentric than isometric contractions. Huxley (1957) conducted a study of muscle
at a microscopic scale and developed a theory concerning the structure and action
of muscle at the molecular level, and developed the cross-bridge muscle model.
Gordon et al. (1964, 1966a, b) continued this work and described this force-length
relationship.
It has been shown that the Hill model of muscle action does not accurately
represent all the characteristics of muscular contraction. Despite the limitations of
the model it has still become widely accepted and has been used in many
successful simulation models. The primary strengths of Hill-type models is in
their simplicity and their ability to represent the function of a whole muscle in
vivo; because so little computation is required, multiple Hill-type models can be
used within the same simulation. Anderson and Pandy (1999) employed a 10-
segment model driven by 54 different Hill-type models to simulate vertical
jumping in three dimensions.
Sporting simulations, such as Anderson and Pandy (1999), have used whole body
models driven by many muscle models. However this adds an unnecessary level
of complexity unless the sequencing of individual muscle activity is the question
at hand. Recent studies have moved toward the use of a single torque generator
about a joint in order to represent the rotational effects of all the muscles acting
about the joint (Yeadon & King, 2002).
27
2.5.1.2 Strength Measurement
In order to personalise a simulation model to an individual, muscle parameters
must be based on the capabilities of that person; this requires knowledge of the
strength of the individual in performing particular actions. It has been shown
previously in experimental studies that measured EMG activity can be used as a
measure of the torque produced about a joint (Cramer, 2002; Tate & Damiano,
2002) and a relationship between EMG and muscle force has long been suggested
(Bayer & Flechtenmacher, 1950). However the exact nature of the relationship
and the mechanism that acts between EMG and muscle force is still disputed and
so the ultimate goal of determining muscle forces from EMG data has been
attempted using quantitative data (Hof & van den Berg, 1981a, b, c, d) but cannot
yet be performed reliably; Alkner et al. (2000) found three differing relationships
for three different quadriceps muscles in a single subject. It seems that the
relationship between EMG and muscle force remains uncertain and does not
provide an accurate means to drive a simulation model.
A single torque generator can be used to represent the net torque produced by all
the muscles that act about a joint removing the complexities of individual muscle
actions, and the torque production capabilities of an individual can be measured
directly using an isovelocity dynamometer. Electronic isovelocity dynamometers
(e.g. Contrex, Biodex, Kin-Com) measure the torque applied by a muscle group
about a joint during isotonic concentric and eccentric motions, as well as during
isometric contractions. Data can be collected for all the required muscle groups
for a large range of velocities and can be processed to provide a complete strength
profile for a subject to be used in conjunction with a simulation model (King &
Yeadon, 2002; Yeadon et al, 2006a).
An isovelocity dynamometer measures the net torque applied to a mechanical
lever arm rotating about an axis therefore the axes of the joint and the
dynamometer must be aligned properly and fixed securely in place or a
conversion must take place so the torque data corresponds to the actual joint angle
rather than the crank angle of the dynamometer. The net torque output of the
dynamometer must also be corrected for the torque due to gravity (Herzog, 1988)
as this has been shown to introduce up to 510% error in knee flexion (Winter et
28
al., 1981). Herzog (1988) also noted that the inertial effects of the machine and
the non-rigidity of the crank arm-human system must be taken into account in
order to accurately derive joint torque data from the dynamometer output.
As discussed earlier, human muscle is complex and accurate representation of its
properties requires consideration of the velocity and length of the muscle. In order
to create an accurate representation of an individual’s ability to create a torque
about a joint the isovelocity dynamometer must be used to collect data over a
range of velocities and joint angles that encompasses the joint motions that the
simulation model will be applied to. Kawakami et al. (2002) studied the shift of
the angle at which peak torque occurred with velocity and concluded that the
evaluation of force-velocity relationship of muscle through isovelocity
dynamometry should be conducted by means of peak torque rather than angle-
specific torque. On the other hand Rácz et al. (2002) concluded that the use of
mean torque reflected the working capacity of the muscle and allowed a superior
fit to a Hill-type model. However, in order to represent all the properties of
muscle a three-dimensional surface function must be adopted to show the
maximum torque capabilities at a particular angle and velocity (e.g. Khalaf et al.,
2000; King & Yeadon, 2002).
Such torque profiles model maximal torques and assume that the torque
production about a joint is only dependent on the position and velocity of the joint
but this is not the case, the torque produced about a joint is also dependent on the
activation level of the muscles that act about the joint (Westing et al., 1990) and,
in the case of biarticular muscles, can also depend on the position and velocity of
adjacent joints.
Yeadon et al. (2006a) combined a model of a tetanic torque-angular velocity
relationship with an activation-angular velocity relationship and the resulting
product was able to closely fit experimental data, the study concluded that an
activation profile must be included in a model simulating actions with both
maximal concentric and eccentric phases. Starting knee angle has also been shown
to affect the angle of peak torque (Pavol & Grabiner, 2000); this is possibly
associated with activation levels. However this highlights the importance of the
range over which torque is measured and movement history; torques measured
29
over a range of angles should not be applied to movements over angles outside the
measured range without caution and previous movements should be taken into
consideration.
Pavol and Grabiner (2000) also found that both the knee and hip angles had large
effects on knee extensor torques as well as there being a large variation in the
effects between subjects. Attempts have been made to understand the
contributions of biarticular muscles to various motions in investigations utilising
simulation models (e.g. Jacobs et al., 1996) and it would be advantageous to
include the effects of adjacent joint positions and velocities within the model of a
torque generator that represents the action of biarticular muscles. Lewis et al.
(2012) modelled the leg and found that whilst torque-driven models that account
for the angle and angular velocity of adjacent joints can more accurately represent
the torque output of biarticular muscles, it is only necessary when the knee was
flexed by an angle of 40° or greater.
Previous muscle-driven simulation models have obtained their muscle parameters
from values published in various literature sources, however such models are not
specific to any particular subject and therefore the validity of their application is
limited. Attempts have also been made to construct a generic torque-driven model
from data collected from a small population, although it was found that inter-
subject variation in torque-velocity responses greatly limited the value of such a
model (Hawkins & Smeulders, 1999).
King and Yeadon (2002) developed a method of determining a set of subject-
specific strength parameters to be used in conjunction with a simulation model of
dynamic jumping by simulating the muscle function of contractile and series
elastic components. An eighteen parameter surface function was produced to
express the torque-angle-angular velocity relationship of a joint in which the
torque-angular velocity relationship was modelled by a six parameter exponential
function. Each of the positive parameters was expressed as a quadratic function of
joint angle to incorporate the torque-angle relationship. The model parameters
were determined by fitting the relationship to data obtained from experimental
protocol using an isovelocity dynamometer. Wilson (2003) developed a nine
parameter function using a similar method.
30
Yeadon et al. (2006) combined a seven parameter function that matched torque-
angular velocity profiles measured using an isovelocity dynamometer in
conjunction with a three parameter activation profile. The inclusion of the
activation profile improved the specificity of the muscle-tendon complex to the
function of human muscle. The activation profile incorporated the ramping
characteristics of human muscle activation. Such activation profiles can be used to
control the number of times that a torque generator can ramp up and down.
Recent methods have used indirect measurements of joint torques; King et al.
(2009) developed a novel method to determine joint torque parameters without
direct measurement. Joint torques profiles were obtained from computer
simulations of multiple actual diving takeoff performances, with the maximum
values taken from the selection of performances. This meant that the final joint
torque profiles could achieve the maximum torques produced during the recorded
performances. This approach has the advantage of not requiring dynamometer
measurements to implement, which also requires both access to the subject and
time to measure and process the data.
2.5.1.3 Summary
In order for a simulation model to represent human performance accurately it
must be limited in its abilities by parameters that are the equivalent of the limiting
factors in humans. One of the primary limiting factors in sporting performance is
strength, which can be measured using isovelocity dynamometers over a range of
velocities and angles at a number of joints. The resulting strength profiles can then
be used in conjunction with muscle models to power whole-body simulation
models using a single torque generator at each joint, or contractile elements
representing individual muscles, or simply using the strength profile to limit the
torques produced by an angle-driven model.
2.5.2 Body Segmental Inertia Parameters
In order to simulate the motion of a body it is essential to know the inertial
characteristics of the body and all its constituent parts. These inertial
characteristics are the mass, position of the centre of mass and the moment of
31
inertia, and when simulating a movement using a model consisting of multiple
segments it is necessary to have knowledge of these three parameters for each
segment of the simulation model. Previous studies have obtained this data for
human beings from dissecting cadavers and directly measuring each quantity
experimentally (Chandler et al., 1975; Dempster, 1955) and the resulting
segmental inertia parameter data has been used in many different simulation
models (e.g. Yeadon, 1990c).
Cadaver data can also be scaled by use of regression equations (Hinrichs, 1985;
Yeadon & Morlock, 1989; Zatsiorsky & Seluyanov, 1985) and other methods
(Forwood et al., 1985). Forwood et al. (1985) found scaling to height to be better
than scaling to segment lengths although this was still inaccurate and the method
was not applicable about the longitudinal axes of segments. Hinrichs (1985) was
successful in scaling inertia parameters using linear regression equations but only
for a limited range of statures. In the process of reducing the data for analysis each
segment was simplified to be symmetrical about the longitudinal axis. Zatsiorsky
and Seluyanov (1985) developed predictive regression equations based on cadaver
data and data obtained from CT scans (Zatsiorsky & Seluyanov, 1983), the
reference points for which were then adjusted by de Leva (1996) to enable them to
be used more easily. Yeadon and Morlock (1989) investigated the use of different
linear and non-linear regression equations to estimate inertia parameters based on
the data of Chandler et al. (1975), and discovered that non-linear regressions were
able to predict moments of inertia with less than twenty percent error whereas
other regression equation could give negative moments of inertia in extreme body
types (Hinrichs, 1985). However, although Chandler’s data can be scaled to
individuals, the cadavers from which the data were taken were not representative
of an athletic population and so the scaled data may not even be close to the actual
inertial characteristics of an athlete, and a set of subject-specific segmental inertia
parameters would be a large improvement.
Subject-specific segmental inertia parameters can be obtained from geometrical
models such as those developed by Hanavan (1964), Jensen (1976), Hatze (1980)
and Yeadon (1990). Geometrical models represent the body as a number of
different geometric shapes in place of the body segments, the dimensions of the
32
shapes are scaled to fit anthropometric measurements taken from the subject and
are given densities based on cadaver data. From the known shapes and densities
the segmental inertia parameters can then be calculated for use in a simulation
model. Such geometric models differ from each other in the different segments
used in the model, the shapes that are used to represent the segments and the
methods for obtaining the anthropometric measurements.
Sarfaty and Ladin (1993) developed a method of estimating segmental inertia
parameters based on video footage and density data from Dempster (1955). The
method modelled segments as cylinders and assumed that mass was evenly
distributed throughout the body segments.
Done and Quesada (2006) used an innovative method of deriving body segment
parameters for the lower leg from kinematic data and evaluating work and energy
of the body segment. Durkin and Dowling (2006) experimented with using dual
energy X-ray absorptiometry (DEXA) scans and anthropometric measurements to
develop and validated a three-segment model of the lower leg; however this
method would be expensive to determine a complete set of inertia parameters for
multiple subjects.
The various methods of estimating body segment parameters have been compared
within experimental simulations of complex airborne movements by Kwon (1996;
2000; 2001). Kwon (2001) found that methods provided very similar results for
rotations about the lateral axis although body segment parameters determined
through personalised methods provided more accurate rotations about the
longitudinal and frontal axes. Hatze (2005) also compared different methods of
estimating segmental inertias and concluded that the use of anthropometric
measurements was the most accurate method.
2.5.3 Kinematic Data
2.5.3.1 Image Analysis
Alongside the measurement of forces and other kinetic methods, the analysis of
images to obtain kinematic data is a primary branch of biomechanics. Two-
dimensional image analysis has been widely carried out through various methods
33
of digitisation where the positions of landmarks of the image are logged as
coordinates. However more recently analysis of three-dimensional motion has
become increasingly important and so three-dimensional analysis of images is
required. Three-dimensional image analysis requires movements to be filmed
within a calibrated environment by two or more cameras for digitisation by
manual or automatic systems. The images must be synchronised so that the timing
offsets between all images are known. This can be achieved through the use of
timing lights, genlocking the cameras, observation of critical events in explosive
movements, through mathematical methods (Pourcelot et al., 2000; Yeadon &
King, 1999) or by using the audio band (Leite de Barros et al., 2006). The
synchronised and digitised image coordinates can then be reconstructed in three-
dimensional space using the direct linear transformation (DLT) (Abdel-Aziz &
Karara, 1971). DLT requires 11 parameters, concerning the position and
orientation of the camera, and scale and shear factors of the images, to be
calculated from points in the calibration images before the movement coordinates
can be reconstructed in three dimensions. These points used to calculate the
parameters should be spread evenly throughout the calibration volume to achieve
accuracy through the space (Yeadon & Challis, 1994). DLT methods have also
been developed to accommodate panning (Yu et al., 1993) and tilting has been
accomplished in a non-DLT method (Yeadon, 1989) of cameras for use over
larger areas or where space is limited.
2.5.3.2 Data Processing
Raw kinematic data obtained from digitisation, whether it be manual or automatic,
will contain random noise created by errors in the digitisation process that
contaminate the signal. Random noise can be amplified by any numerical
differentiations that may be performed on the data. Ideally there would be no
noise and to try and remove the noise, and increase the signal-to-noise ratio of the
data, smoothing procedures, in the form of mathematical functions, can be
applied. Butterworth filters (Pezzack et al., 1977), Fourier series (Hatze, 1981b),
quintic splines (Wood & Jennings, 1979) and quintic splines with cross-validation
(Craven & Wahba, 1979; Woltring, 1985) are some of the methods of data
smoothing. However the amount of smoothing required and other effects of
34
smoothing procedures must be considered in order to select an appropriate method
(Yeadon & Challis, 1994). All the methods mentioned above are proven to fit
displacement data accurately although a digital filter such as a Butterworth filter
comprises two frequency cut off points outside of which data is removed, this
results in an output that is not suitable for further computations such as
differentiation. Other functions, such as Fourier series and cross-validated
polynomial fits require equidistant data and so can only be applied to data with
consistent sampling rates (Wood, 1982; Woltring, 1985). Wood (1982) concluded
that spline functions were ideal to interpolate time history data, whilst quintic
splines have been shown to provide the most accurate second derivative data
(Challis & Kerwin, 1988) and not to suffer from boundary effects in the endpoint
regions (Vint & Hinrichs, 1996; Woltring, 1985).
2.6 Optimisation
Optimisation algorithms are frequently used alongside simulation models to
search for an optimal solution to a problem, to find an optimal technique (e.g.
Hiley & Yeadon, 2007) or to match a simulation to a performance to attain the
best set of parameter values (e.g. Ait-Haddou et al., 2004). The optimisation of
computer simulations is necessary when there are parameters that govern the
performance that cannot be measured, either directly or indirectly. Search
algorithms are able to converge on the optimal set of parameters through the
systematic search of a specified area to minimise or maximise a cost function.
The optimisation of sports techniques has been accomplished using various
different optimisation algorithms including the simulated annealing algorithm
(Corana et al., 1987), Powell’s algorithm (Press et al., 1992) and the downhill
simplex method (Press, 1997) and genetic algorithms (Davis, 1991). Goffe et al.
(1994) found the simulated annealing method (Corana et al., 1987) to be superior
to other optimisation procedures for it is able to find a global optimum rather than
local optima as well as being robust to exceptionally difficult problems. Whilst
van Soest and Casius (2003) found that both simulated annealing and genetic
35
algorithms were able to find a global optimum solution when solving tough
optimisation problems.
Simulated annealing is a heuristic search algorithm which learns about its search
space with each solution and takes a probabilistic approach to determining its next
starting point. It is based on the cooling of a material to form crystals by lowering
the total energy within the system, taking the form of the cost function. The speed
of the convergence is governed to allow the search to find the global optimal
solution and not get stuck in local optima (Corana et al., 1987).
Genetic algorithms solve a problem by modelling the problem as a population
undergoing evolution via natural selection. Chromosomes representing the
parameters from the population are selectively mated with one another depending
on their fitness for purpose, evolving the population through generations until a
solution is found (Davis, 1991). However the genetic algorithm will take a long
time to find the best solution to a problem if the population size is too large (Harik
et al., 1999).
In the past computational limitations has caused inferior optimisation methods to
be used for the sake of saving time, however as computational power has
improved the time taken to run optimisations has been dramatically reduced and
more complex problems have been optimised (e.g. Anderson & Pandy, 1999).
Optimisation algorithms can also perform very differently when asked to solve
different kind of problems, simple functions without local optima can be solved
quickly and reliably using a simple downhill method, however more complex
functions, like those describing human movements, require a more robust
approach.
2.7 Chapter Summary
Torque-driven simulation models can be personalised to a subject by using a set of
subject-specific strength parameters determined experimentally on an isovelocity
dynamometer measuring the net torque produced about specific joints. Body
segmental inertia parameters can be accurately calculated from anthropometric
36
measurements used as input for a geometric model. Kinematic data describing the
motions performed to be simulated by the model can be obtained by smoothing
raw data collected through the digitisation and reconstruction of synchronised
video images into three-dimensional coordinates. The next chapter describes the
collection and processing of kinematic and anthropometric data.
37
Chapter 3: Data Collection
3.1 Chapter Overview
In this chapter the experimental protocol to collect kinematic and anthropometric
data from a trampolinist is described. The methods used to process the kinematic
data are detailed. Segmental inertia parameters for the trampolinist are reported.
3.2 Kinematic Data Collection
3.2.1 Camera Set-up
A VICON motion capture system, comprising 16 MX cameras, was used to record
trampolining performances on a sunken trampoline. The motion capture system
was set-up to capture a volume extending to five metres above and one metre
below the level of the trampoline bed but with particular focus on capturing the
right-hand side of the subject. The positions of the cameras are shown in Figures
3.2.1 and 3.2.2. The volume was calibrated so that each camera was accurate to
within 0.35 mm and gave a dynamic wand test accuracy of 0.004%. The system
was calibrated at 480 Hz with kinematic data subsequently recorded at 300 Hz.
38
Figure 3.2.1 A view of the experimental set up.
Figure 3.2.2 Illustration of the 16 camera positions for data collection.
3.2.2 Data Collection
An elite male trampolinist competing at junior international level (mass = 60.1 kg,
height = 1.69 m) participated in the study. The subject was briefed on the data
collection procedure before written informed consent was obtained (Appendix 1).
Opto-reflective markers were placed on landmarks of the right-hand side of the
trampoline pit area
39
body as well as the left hip and head. The full marker set comprised markers on
the front and back of the head, the sternum, both hips, the posterior aspect of the
right shoulder, elbow, wrist, the lateral aspect of the right knee and ankle, and on
the superior aspect of the metatarsal-phalangeal joint and toes of the right foot as
illustrated in Figure 3.2.3. The markers were placed over the joint centres and
landmarks, so in order to relate the marker positions to the joint centre locations
and to help reflect the position coordinates of the right-side of the body to
represent the left-side, a number of marker offsets were measured. The offset
measurements included the position of the foot markers in relation to the floor,
metatarsal-phalangeal joint centre and midline of the toe, and the lateral distance
between the markers on the limbs and the midline; a full list of offsets can be
found in Table 3.2.1. The values of the measured offsets can be found in
Appendix 2b.
Figure 3.2.3 Photograph showing the marker set employed to collect position data.
40
Table 3.2.1 List of marker offsets measured.
Marker Offset axes
Sternum frontal
Shoulder lateral
Elbow lateral
Wrist lateral
Hip lateral
Ankle lateral, frontal
Ball frontal
Toe frontal
Following the completion of the measurements the subject was given a period of
familiarisation with the trampoline before performing a specified sequence of
forward and backward rotating skills with various amounts of somersault as well
as straight jumps of different heights. Each skill was performed until a satisfactory
trial with no noticeable travel or cast was obtained. The trampolinist performed 21
skills in total in order to obtain satisfactory trials for 12 different skills (see Table
3.2.2).
41
Table 3.2.2 List of recorded performances.
Skill Shape Satisfactory?
Front S/S Straight Y
1¾ Front S/S Pike Y
1¾ Front S/S Open Pike N
1¾ Front S/S Open Pike Y
2¾ Front S/S Tuck Y
Triffus Pike N
Triffus Pike Y
High Jumping Y
Back S/S Pike N
Back S/S Pike N
Back S/S Pike Y
Back S/S Straight N
Back S/S Straight Y
1¼ Back S/S Straight N
1¼ Back S/S Straight Y
Double Back S/S Tuck N
Double Back S/S Tuck Y
Double Back S/S Pike N
Double Back S/S Pike N
Double Back S/S Pike Y
Medium Jumping Y
3.2.3 Body Segmental Inertia Parameters
Body segmental inertia parameters of the trampolinist were calculated using 97
anthropometric measurements in conjunction with the mathematical inertia model
of Yeadon (1990b) (Appendix 2). Table 3.2.3 shows the mass, moment of inertia
42
and length of each segment as well as the distance of the centre of mass of the
segment from the proximal joint. For the limbs the mass and moment of inertia
included are those of the combined left and right limbs with the average length
and distance of the centre of mass from the proximal joint given.
Table 3.2.3 Body segmental inertia parameters calculated using the inertia model of Yeadon
(1990b).
Segment Mass Length Moment of Inertia CM distance from
proximal joint
(kg) (m) (kg.m2) (m)
forearm and hand 3.13 0.426 0.041 0.162
upper arm 3.28 0.245 0.009 0.112
trunk, head and neck 29.13 0.865 1.554 0.383
thigh 14.91 0.372 0.178 0.165
lower leg 7.76 0.407 0.100 0.171
foot 1.59 0.145 0.003 0.061
toes 0.28 0.065 0.0002 0.028
The foot was modelled as two segments: a triangle representing the foot between
the ankle and metatarsal-phalangeal joints, and a rod representing the toes, shown
in Figure 3.2.4. The extra segments required additional segmental inertia
parameters describing the position of the centre of mass within each segment, the
mass and moment of inertia of each individual segment.
43
Figure 3.2.4 The foot modelled by a triangle and rod segment.
Lengths L1-L4 were calculated by the inertia model of Yeadon (1990b), L6 and
L7 were measured directly from the subject during the data collection, whilst L5
was estimated using a subject-specific scaling method. Table 3.2.4 summarises the
dimensions of the two-segment foot.
Table 3.2.4 Dimensions of the two-segment foot.
Parameter Length
(mm)
L1 65
L2 28
L3 145
L4 61
L5 17
L6 48
L7 78
44
3.3 Kinematic Data Processing
3.3.1 Joint Centre Positions
The position data recorded by the motion capture system and the offset
measurements were combined, under the assumption of symmetrical body
movement to create a whole-body pseudo-set of three-dimensional joint centre
position data. This involved the relocation of data points towards the joint centres
using the offset measurements and the calculation of a central head position.
Transposing data points along the frontal axis was potentially a problematic
activity as these offsets were within the primary plane of motion. However the
sternal landmark data did not need to be transposed since it was not to be
employed as input to the simulation model and a method used to relocate the
position of the metatarsal-phalangeal joint and the toe centres was devised to
minimise the associated errors.
The method used to transpose the foot landmarks along the frontal axis was to
subtract the offset of the height of the marker centre from the base of the marker
from the offset measurement leaving the depth of the foot at these locations. The
offsets at the metatarsal-phalangeal joint and toe were calculated by adding the
height of the marker centre to half the calculated depth of the foot at these points.
These offsets were applied in the direction perpendicular to the line passing
through the ankle marker and the original marker position, shown in Figure 3.3.1.
Figure 3.3.1 The transposition of foot markers.
45
The offset measurements taken in the direction of the lateral axis were used to
transpose the data positions to the joint centres before the joint centre positions
were reflected in the plane of the body that bisected the two hip markers
vertically, in order to create a full body data set. The position of the centre of the
head was taken to be the mean position of the markers on the front and back of the
head.
The segment lengths calculated from the joint centre position data, shown in Table
3.3.1, are consistent between separate trials as shown by the small standard
deviations, the largest standard deviations being found in the lengths of the head
and trunk segments. Within the trials the standard deviations of the segment
lengths were larger, and once again the segments with the largest variability in
length were the head and trunk.
Table 3.3.1 Mean segment lengths calculated from joint centre position data.
Upper
Arm
Forearm
and Hand Thigh
Lower
Leg Foot Trunk Head
(m) (m) (m) (m) (m) (m) (m)
mean 0.256 0.232 0.447 0.397 0.216 0.436 0.273
standard
deviation
between trials
0.009 0.003 0.016 0.008 0.003 0.023 0.019
mean standard
deviation within
trials
0.018 0.007 0.010 0.005 0.017 0.049 0.053
The increased variability in length within trials could possibly be attributed to
marker movement due to movement of the underlying soft tissue or clothing. The
trunk segment was defined by the shoulder and hip markers. It is possible that the
movements of the shoulder joint could affect the position of the marker even
though the marker position was chosen to minimise this possible effect, the hip
marker may also have been subject to movement as it was placed on top of
46
(3.3.1)
clothing. The head segment also used the shoulder marker as a reference point so
may have also been influenced by shoulder joint movements.
Table 3.3.2 Differences in segment lengths between the inertia model and static trials.
By examining the difference between the segment lengths given by the inertia
model output and the static trials it was observed that there was a difference in
length of 59 mm in the trunk segment and -68 mm in the thigh segment. On
examination of photographs of the subject it seemed that clothing may have been
rearranged following the taking of anthropometric measurements and before the
three dimensional position data was collected, altering the location of the hip
marker. It was estimated that the hip markers were 64 mm higher than the hip
centres, the mean of the differences in trunk and thigh length.
The hip centres calculated from the hip marker positions were adjusted as follows:
𝐻 → 𝐻 +64
565 𝑆𝐻⃗⃗⃗⃗ ⃗ = 𝐻 +
64
565(𝐻 − 𝑆)
Where: H = hip centre location
S = shoulder centre location
Segment Difference
Upper Arm (m) 0.016
Forearm and
Hand (m) -0.014
Thigh (m) 0.068
Lower Leg (m) 0.008
Foot (m) 0.007
Trunk (m) -0.059
Head (m) -0.1
47
3.3.2 Joint Angle Time Histories
The joint centre position data was used to calculate whole body orientation and
joint configuration angles and angular velocities of the gymnast throughout each
of the recorded performances (Yeadon, 1990a). When combined with segmental
inertia parameters, the joint angle time histories allow the calculation of centre of
mass position and velocity, as well as the angular momentum of the whole body
about the centre of mass (Yeadon, 1990c). The joint configuration angles
calculated were those of the metatarsal-phalangeal, ankle, knee, hip, shoulder and
elbow joints; to be used in conjunction with the seven-segment simulation model.
All the joint angles calculated were the interior angles formed between the
adjacent segments in the sagittal plane. The whole body orientation angle was
defined as the angle formed by the shoulder, hip and the forward horizontal. The
time histories of the whole body orientation and joint configuration angles were
fitted using quintic splines (Wood & Jennings, 1979).
48
Figure 3.3.2 Representation of the captured motion of the trampolinist during (a) straight
bouncing, (b) straight front somersault, (c) 1¾ (open) piked front somersault, (d)
2¾ tucked front somersault, (e) triffus piked (before initiating twist).
a.
c.
d.
e.
b.
49
Figure 3.3.3 Representation of the captured motion of the trampolinist during (a) piked back
somersault, (b) straight back somersault, (c) 1¼ straight back somersault, (d) tucked
double back somersault, (e) piked double back somersault.
a.
c.
d.
e.
b.
50
(a) Single front somersault (b) Triple front somersault
(c) Single back somersault (c) Double back somersault
Figure 3.3.4 Joint angle time histories of the knee (solid), hip (dashed), and shoulder (dotted) during
the contact phase directly preceding different somersaults.
The joint angle data was used to inform the design of the simulation model. The
data showed that throughout all the trials there was significant movement of the
elbow joint with a mean range of movement of 46.9±8.1° (forward: 44.8±6.0°,
backward: 45.1±5.6°). The movement of the head and neck was found to be
22.9±10.7° on average, however a larger difference was found between forward
(13.0±3.2°) and backward (31.8±6.9°) somersaults.
The motion of the simulation model was also restricted to joint movements that
the trampolinist was actually able to perform. The data was analysed to find the
limits of the range of motion for each joint movement that the trampolinist
achieved during the data collection procedure. The shoulder showed a full, 360°
range of motion in the recorded performances due to the projection of the
movements on to a two dimensional plane. The minimum and maximum ranges of
motion for the ball, ankle, knee, hip and elbow joints are displayed in Table 3.3.3.
90
135
180
225
0 0.2 0.4Jo
int
An
gle
(°)
Time (s)
90
135
180
225
0 0.2 0.4
Join
t A
ngl
e (°)
Time (s)
90
135
180
225
0 0.2 0.4
Join
t A
ngl
e (°)
Time (s)
90
135
180
225
0 0.2 0.4
Join
t A
ngl
e (°)
Time (s)
51
Table 3.3.3 Minimum and maximum joint angles achieved in the recorded performances.
Joint Limit of Motion (°)
Minimum Maximum
Ball 109.3 151.2
Ankle 95.4 150.3
Knee 135.3 193.7
Hip 108.5 211.1
Elbow 85.3 170.9
Although the hip achieved a large degree of hyperextension, 211.1°, this was
assumed to include some arching of the back and was limited to immediately prior
to takeoff in backward rotating skills. Conversely the knee hyperextension was
limited to immediately prior to takeoff in forward rotating skills.
3.3.3 Trampoline Bed Movement
The movement of the trampoline bed throughout the contact phase of each trial
was also analysed. Each contact was normalised for the maximum depression of
the trampoline bed and the duration of the contact phase for analysis. A quintic
spline was then fit to the data.
Figure 3.3.5 Plot of normalised bed depression against normalised contact time for 15 trials.
z = 1.3057t5 - 11.058t4 + 17.963t3 - 6.1303t2 - 2.074t - 0.021
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1 1.2
No
rma
lise
d B
ed
De
pre
ssio
n
Normalised Contact Duration
52
The point of maximum depression was found to be at 48.7% of the contact
duration with the minimum vertical velocity of the bed occurring 13.6% of the
way through the contact and maximum vertical velocity at 84.2% of the contact
time. Maximum vertical acceleration of the bed occurred just prior to maximum
depression, 47.2% of the way through the contact.
3.4 Chapter Summary
In this chapter the collection of kinematic performance data of trampolining is
described. The techniques for the determination of subject-specific inertia
parameters are described. Details of the data processing techniques were reported.
The following chapter describes the development of a computer simulation model
of a trampolinist and trampoline suspension system.
53
Chapter 4: Model Development
4.1 Chapter Overview
A simulation model representing a trampolinist and the reaction forces of the
trampoline suspension system was required to study the mechanics of the takeoff
from trampoline. This chapter describes the features of the simulation model of
the trampolinist and trampoline suspension system.
4.2 The Trampolinist Model
The trampolinist was represented by a planar seven-segment model consisting of
the torso, head and neck as one segment, the upper arms, lower arms and hands,
thighs, lower legs and a two-segment foot. Each segment was modelled as a single
rigid body as it has been shown that the inclusion of wobbling masses
representing soft tissue movement is not necessary when modelling impacts with
compliant surfaces (Yeadon et al., 2006b). The arm was modelled as two
segments as the data showed significant changes in elbow angles during the
contact phase preceding both forward and backward somersaults. Despite the
range of neck motion preceding backward somersaults being found to be over 30°,
the torso, head, and neck were modelled as a single rigid segment as this
movement was thought to have little mechanical effect. The foot was modelled as
a triangle with a rod, representing the toes, connected at the metatarsal-phalangeal
joint. The orientation and configuration of the trampolinist was described by
seven angles, the angle of the trunk to the vertical θt and the six internal angles at
the shoulder θs, elbow θe, hip θh, knee θk, ankle θa, and metatarsal-phalangeal θm
joints.
54
Figure 4.2.1 A seven-segment angle-driven model of a trampolinist.
4.3 Foot/Suspension-System Interface
During the takeoff from the trampoline, the surface of the trampoline suspension
system is depressed then recoils as the trampolinist lands from the previous flight
phase and jumps into the subsequent flight phase. Previous studies have found the
force-displacement relationships of the trampoline suspension system to be non-
linear vertically and linear horizontally (Blajer & Czaplicki, 2001; Jaques, 2008;
Kraft, 2001; Lephart, 1971), however the exact relationship is individual to each
trampoline.
In the present study, the modelling of the foot-suspension system interface is
simplified to include all the relevant forces within a single expression to represent
the reaction forces acting on the trampolinist. The total reaction force is then
allocated between three points on the foot at the heel, ball and toes. The ratio of
the allocation of reaction force between the three points is based on the depression
of the trampoline suspension system at the heel, ball and toe.
θt
θe
θs
θh
θk
θa
θm
55
(4.3.1)
(4.3.2)
4.3.1 Kinetics of the Foot/Suspension-System Interactions
When a trampolinist lands on the bed of the trampoline suspension system the
reaction force accelerates the mass of the bed and springs downwards, while the
opposite reaction force accelerates the trampolinist upwards. Subsequently, any
elastic force created by the extension of the springs, accelerates the masses of both
the suspension system and the trampolinist upwards.
Figure 4.33.1 Free body diagram showing the vertical forces acting on the trampolinist, G, and
trampoline, B.
𝑅 − 𝑚𝐺𝑔 = 𝑚𝐺 �̈�𝐺
𝐹𝑡𝑜𝑡 − 𝑅 = 𝑚𝐵 �̈�𝐵 ∴ 𝑅 = 𝐹𝑡𝑜𝑡 − 𝑚𝐵�̈�𝐵
Where: 𝑚𝐺 = mass of the gymnast
𝑚𝐵 = mass of the trampoline bed
�̈�𝐺 = vertical acceleration of the gymnast’s mass centre
�̈�𝐵 = vertical acceleration of the trampoline bed’s mass centre
𝐹𝑡𝑜𝑡 = total vertical force exerted by the trampoline suspension system
𝑅 = vertical reaction force acting between the gymnast and
trampoline bed
The weight of the bed of the suspension system has been neglected here as 𝐹𝑡𝑜𝑡
will be calibrated to be zero when the bed is in its stationary position.
56
(4.3.3)
(4.3.4)
For example, if the initial impact of the trampolinist landing on the bed of the
suspension system were to be considered instantaneous, with the stationary bed
with an equivalent mass, 𝑚𝐵 = 1
650 ≈ 8𝑘𝑔 (Jaques, 2008), conservation of linear
momentum would give:
𝑚𝐺𝑣𝐺 = 𝑚𝐺𝑣𝐺′ + 𝑚𝐵𝑣𝐺
′
𝑣𝐺′ =
𝑚𝐺
𝑚𝐺 + 𝑚𝐵𝑣𝐺
Where: 𝑣𝐺= impact velocity of the gymnast
𝑣𝐺′ = velocity of the gymnast after impact
If the mass of the trampolinist, 𝑚𝐺 = 60𝑘𝑔.
𝑣𝐺′ =
60
60 + 8𝑣𝐺 = 0.88𝑣𝐺
Taking an impact velocity, 𝑣𝐺 = −9𝑚𝑠−1 , during the impact the trampolinist
would slow to −7.94𝑚𝑠−1 and the bed of the suspension system would accelerate
to −7.94𝑚𝑠−1.
The impulse, J, of this instantaneous impact is:
𝐽 = 𝑚𝐵𝑚𝐺
𝑚𝐺 + 𝑚𝐵∙ 𝑣𝐺
For example the impulse on the suspension system may be expressed as:
𝑚𝐵𝑣𝐺′ = 8(−7.94) = −63.5𝑁𝑠
Or the impulse on the gymnast as:
𝑚𝐺(𝑣𝐺′ − 𝑣𝐺) = 60(1.06) = 63.5𝑁𝑠
Since the impact is not instantaneous, this impulse is spread over a finite time.
Initially, upon impact, the area of the bed beneath the point of impact will be
accelerated to match the velocity of the feet. This acceleration will occur over a
duration of around 30 ms, the rest of the bed will then be accelerated over a period
57
(4.3.5)
(4.3.6)
of around 100 ms. These estimates were made from inspection of the vertical
locations of the foot and trampoline bed during a landing. The impulse can be
represented by an inverted cosine function as:
𝐹𝑖𝑚𝑝 =𝐽
𝑇[ 1 − 𝑐𝑜𝑠
2𝜋𝑡
𝑇]
Where 𝐹𝑖𝑚𝑝 is the force arising from the impact with the stationary bed of the
suspension system and the impulse J is calculated using Equation 4.3.4.
Figure 4.3.2 Graph showing the development of impact force as a function of T.
In order to determine T using optimisation the bounds for T should be set between
0.06 and 0.18 s as the total time taken for the bed to reach maximum depression is
approximately 180 ms. Fimp provides the impulse required to accelerate the mass
of the trampoline suspension system to match the speed of the feet of the
trampolinist and so should vary with the mass centre vertical velocity of the
trampolinist at the time of contact.
The vertical reaction force R acting on the feet of the trampolinist may be
calculated as:
𝑅 = 𝐹𝑡𝑜𝑡 + 𝐹𝑖𝑚𝑝 − 𝑚𝐵 �̈�𝐵
The acceleration of the bed of the suspension system will not be measured but in a
simulation �̈�𝐵 will be calculated as the foot acceleration, �̈�𝐹. Typically there is no
net change in foot velocity in the first 60𝑚𝑠 of contact; this means that only the
spring force and the impact force are involved in the early portion of the contact
phase via Equation 4.2.6. Foot acceleration can be calculated as an average of the
T 0
Fimp
time
58
(4.3.7)
(4.3.8)
accelerations of the 3 points on the foot with weightings proportional to the
vertical components of 𝐹𝑡𝑜𝑡 or 𝐹𝑡𝑜𝑡 + 𝐹𝑖𝑚𝑝.
For momentum calculations, the movement of the suspension system can be
represented by a mass 𝑚𝐵 moving at 1
6𝑣𝐵 or a mass
1
6𝑚𝐵 moving at 𝑣𝐵, however
when it comes to calculating the energy left in the bed after takeoff it does make a
difference, as 1
2 𝑚𝐵(
1
6𝑣𝐵)2 ≠
1
2 (
1
6𝑚𝐵)𝑣𝐵
2 . If at takeoff the velocity of the bed
of the trampoline suspension system is 4.5𝑚𝑠−1 the energy remaining in the bed
is either
𝐸 = 1
2𝑚𝐵(
1
6𝑣𝐵)2 =
1
250(0.75)2 = 14 𝐽𝑜𝑢𝑙𝑒𝑠
or
𝐸 = 1
2(1
6𝑚𝐵)𝑣𝐵
2 =1
28(4.5)2 = 84 𝐽𝑜𝑢𝑙𝑒𝑠
If the trampolinist takes off at 9𝑚𝑠−1 then this equates to 0.5% or 3.5% of the
energy being lost in the suspension system. In reality the amount of energy lost is
likely to lie between these two extreme values, possibly around 2%.
Energy will also be lost due to the viscosity of the suspension system. To
incorporate this energy loss into the modelling of the foot-suspension system
interactions, a damping force, proportional to the square of the bed velocity:
𝐹𝑑 = 𝑘𝑣𝐵2
will act between the foot and the suspension system.
The total force acting on the foot of the simulation model of the trampolinist will
take the form:
𝑅 = 𝐹𝑡𝑜𝑡 + 𝐹𝑖𝑚𝑝 − 𝑚𝐵 �̈�𝐵 − 𝑘𝑣𝐵|𝑣𝐵|
The vertical and horizontal force-displacement relationships of the trampoline
suspension system will be represented by non-linear and linear relationships
respectively (Jaques, 2008):
59
(4.3.9)
(4.3.10)
𝐹𝑍𝑡𝑜𝑡 = 𝑘1𝑧2 + 𝑘2𝑧
Where: 𝐹𝑍𝑡𝑜𝑡 = vertical reaction force
k1 = non-linear vertical spring stiffness
k2 = linear vertical spring stiffness
z = vertical displacement
𝐹𝑌𝑡𝑜𝑡 = 𝑘𝑦
Where: 𝐹𝑌𝑡𝑜𝑡 = horizontal reaction force
k = horizontal spring stiffness
y = horizontal displacement
The spring stiffness and damping parameters were determined, along with
parameters describing the effective mass of the trampoline and impact force
period, through a matching procedure in the evaluation of the simulation model.
4.3.2 Locating the Centre of Pressure during Foot Contact
The reaction force described is allocated between three points on the foot in ratios
governed by the relative depressions of the three points in order to accurately
represent the position of the centre of pressure beneath the foot on the compliant
surface. The surface of the trampoline suspension system acts in such a way that
when depressed the surface of the suspension system forms new contours and the
suspensions system will only exert reaction forces around a localised depression if
an adjacent point is depressed beyond the new contours of the suspension system.
In the following description it is assumed that the total vertical force F, is a non-
linear function of the maximal vertical depression of the trampoline surface,
corresponding to the lowest part of the foot. Within the simulation model the total
vertical reaction force acting on the foot F, is represented by three reaction forces
exerted at the heel H, ball B, and toe T.
60
(4.3.11)
(4.3.12)
Figure 4.3.3 Showing the three separate reaction forces acting on the foot.
Suppose that the ratio of the distances 𝑑, between the three points is known:
𝑑(𝐻𝐵) = 0.7 𝑑(𝐻𝑇) and 𝑑(𝐵𝑇) = 0.3 𝑑(𝐻𝑇)
If we assume that the natural ‘slope’ (sinΦ) of the bed is kz, where z is the
maximum depression of the surface and k is a constant. The angle of the foot (HB)
and toe (BT) segments as proportions of the natural slope can be expressed as µ
and λ respectively.
µ =(𝑧𝐻−𝑧𝐵)
𝑑(𝐻𝐵)𝑘𝑧 where −1 ≤ µ ≤ 1
If μ < -1 then the ball is unloaded (FB = 0), so let μ = -1. If μ > 1 then the heel is
unloaded (FH = 0), so let μ = 1.
Similarly,
λ =(𝑧𝑇−𝑧𝐵)
𝑑(𝐵𝑇)𝑘𝑧 where −1 ≤ λ ≤ 1
If λ < -1 then the toe is unloaded (FT = 0), so let λ = -1. If λ > 1 then the ball is
unloaded (FB = 0), so let λ = 1.
FH
FB FT
61
(4.3.13)
(4.3.14)
(4.3.15)
(4.2.13)
(4.3.16)
(4.3.17)
(4.3.18)
(4.2.13)
If H, B and T are at the same level then µ = 𝜆 = 0 and the pressure distribution is
even along the length of the foot. This may be represented by:
𝐹𝐻 = 0.35 𝐹
𝐹𝐵 = 0.35 𝐹 + 0.15 𝐹 = 0.50 𝐹
𝐹𝑇 = 0.15 𝐹
which has a centre of pressure at the midpoint between H and T, where 𝐹 is the
total force.
If H, B and T are at different levels then we may define the vertical forces as
follows.
𝐹𝐻 = [0.35 µ′ − (0.15 λ × 0.5 µ′)] 𝐹
𝐹𝐵 = [0.5 + (0.35µ × 0.5 λ′) − (0.15 λ × 0.5 µ′)] 𝐹
𝐹𝑇 = [0.15 λ′ − (0.35 µ × 0.5 λ′)] 𝐹
Where µ′ = 1 − µ and λ′ = 1 − λ.
So that the sum of FH, FB, and FT is F, and when µ = λ = 0 Equations 4.3.16-18
give the relationships described by (4.3.13-15).
This process allocates the proportion of the total vertical force over the three
points based on the values of µ and λ, representing the slope of the bed around the
point of maximal depression whilst maintaining a vertical force at a minimum of
one point.
If λ = ±1 and µ = ±1, then all the vertical force is exerted at the lowest point.
If µ = 1, then the centre of pressure lies between B and T and moves towards the
lowest as λ → ±1.
62
4.3.3 Determining the Natural Slope of the Surface of the Suspension
System during Depression
In order to allocate accurately the reaction force from the trampoline between
multiple points on the foot, we must be able to say how much further a second,
adjacent point is depressing the trampoline given another point of greater
depression.
The natural slope, kz, of the trampoline from a depressed point was measured
from video of a single contact phase. The position of the toe was digitised in each
frame, along with a point on the surface of the suspension system consistent with
the initial slope of the bed moving away from the toe.
Figure 4.3.4 The points used to determine the natural slope of the suspension system.
The natural slope, defined as Δz/Δy, of the trampoline bed was then calculated
from the digitised points in each frame and the relationship between the natural
slope and the depression of the trampoline bed was determined. A linear function
was fitted to the data and a relationship between the natural slope of the surface of
the suspension system and the point of maximal depression was found.
63
Figure 4.3.5 Graph showing the relationship between natural slope and the depression of the
surface of the suspension system.
The relationship determined between the natural slope of the surface of the
suspension system and the depression of the trampoline was used as the initial
value but was allowed to vary in the optimisation of the model parameters.
4.3.4 Equations of Motion
The equations of motion were formulated for both the angle-driven and torque-
driven models using AutolevTM
Professional Version 4.1. The software package
facilitates the creation of multibody simulation models and uses Kane’s method
(Kane and Levinson, 1996) to derive the equations of motion. Kane’s method uses
generalised coordinates for each segment to define the position and orientation
relative to the global coordinate system or other, previously defined segments.
Generalised speeds are calculated as the time derivatives of the generalised
coordinates (Kane and Levinson, 1985). Inertia parameters, torques, and internal
and external forces are defined so that expressions for the generalised inertia and
generalised active forces can be calculated. The AutolevTM
command files
(Appendix 3), when run produced an output code in FORTRAN programming
language, containing the equations of motion derived using Kane’s method and
code to advance the simulation over a period of time using a Kutta – Merson
numerical integration algorithm with a variable step size Runge-Kutta integration
method. The FORTRAN programs were customised to create the foot-suspension
system interface described in this chapter and to incorporate the torque generators
as described in Section 6.3.
natural slope = 1.226*depression
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Na
tura
l S
lop
e
Depression (m)
64
4.4 Chapter Summary
In this chapter the details of the construction of the computer simulation model
have been described. The force-displacement relationships governing the
interactions between the models of the trampolinist and the trampoline suspension
system were developed and the methods used to determine the distribution of the
reaction force on the feet of the simulation model was described. The following
chapter will detail the application of the angle-driven simulation model of
trampolining.
65
Chapter 5: The Angle-Driven Model
5.1 Chapter Overview
The present chapter describes the method for the determination of the visco-elastic
properties of the trampoline suspension system using an angle-driven simulation
model. The results from the determination of the visco-elastic parameters are
detailed. The method for the evaluation of the angle-driven model is then detailed
and the results are reported.
5.2 Visco-Elastic Parameter Determination
The simulation model requires the input of numerous parameters in order for the
resulting simulations to accurately mimic the movements and interactions of a
trampolinist and trampoline suspension system. Some of these parameters can be
measured directly, whilst others, for example the elastic properties of the
trampoline during impact, are very difficult to measure directly through
experimental methods. In order to determine values for the uncertain parameters, a
subject-specific, angle-driven model was developed.
The angle-driven model was used to ascertain values for the eight parameters that
govern the modelled force interactions of the foot and the trampoline suspension.
These parameters are three spring stiffness and two damping parameters
describing the vertical and horizontal force-displacement relationships, along with
parameters describing the effective mass of the trampoline, impact force period
and natural slope of the trampoline.
By using known initial conditions and driving the model with joint angle time
histories from recorded performances, an optimisation procedure could be used to
determine the uncertain parameters by letting them vary whilst minimising the
difference between the simulation and performance. A simulated annealing
algorithm (Corana et al., 1987) varied the model parameters governing the foot-
suspension system interactions to minimise a cost function designed to match
66
simulations to recorded performances. In order to obtain results from the matching
procedure that can be used to simulate a varied range of different skills a number
of different performances were used (Wilson et al., 2006). Seven trials were
selected to be used in the matching procedure. These trials covered a broad range
of rotational requirements over both forward and backward somersaults; the skills
were: straight jumping S, a straight single forward somersault F1, a piked 1¾
forward somersault F2, a piked triffus F3, a piked single backward somersault B1,
a straight single backward somersault B2 and a piked double backward somersault
B3.
5.2.1 Model Inputs
The inputs of the angle-driven model were the initial conditions of the
trampolinist immediately prior to first contact with the trampoline and body
segmental inertias. The initial conditions prior to contact were the horizontal and
vertical position and velocities of the toe, orientation angle and angular velocity of
the trunk. Throughout a simulation the movements of the simulation model were
driven by joint angle time histories. Both the initial conditions and joint angle
time histories were determined from the two dimensional pseudo-data sets
generated from the kinematic data. The output of the model included the
horizontal and vertical mass centre velocities, trunk angle and whole–body
angular momentum.
67
Table 5.2.1 Initial conditions of seven selected skills.
Skill
Initial condition S F1 F2 F3 B1 B2 B3
Hor. toe
position (m) -0.405 -0.226 -0.364 0.106 -0.197 0.046 -0.031
Vert. toe
position (m) 0.119 0.090 0.101 0.076 0.087 0.096 0.081
Hor. toe
velocity (ms
-1) 0.858 0.155 0.912 0.375 0.146 -0.183 -0.236
Vert. toe
velocity (ms
-1) -7.83 -7.51 -7.24 -7.43 -7.79 -7.52 -7.32
Orientation (°) 72.9 77.2 77.3 76.5 77.5 77.9 77.0
Trunk ang.
velocity (°s
-1) 40.0 -13.3 -32.9 -15.4 -30.4 -34.4 -11.1
During the matching procedure the initial horizontal and vertical velocities of the
toe, and angular velocity were allowed to vary slightly for each individual trial so
that the simulated initial centre of mass velocities could be adjusted to match to
the recorded centre of mass velocity. The horizontal and vertical velocities of the
toe were allowed to vary by up to 1 ms-1
and the angular velocity was allowed to
vary by up to 0.5 rad. s-1
.
5.2.2 Cost Function
A cost function was developed in order to assess objectively the extent to which
each simulation matched the recorded performances. The cost function was
designed so that the objective score it outputted, when minimised, would match
the movement characteristics of the simulation to the recorded performance both
throughout the contact phase and at takeoff.
The cost function comprised terms comparing the initial horizontal centre of mass
position, initial horizontal and vertical centre of mass velocities, toe position in
both the horizontal and vertical dimensions throughout the simulation, horizontal
and vertical takeoff velocities, orientation angle and angular velocity of the trunk.
68
(5.2.1)
The toe position difference was calculated as a root mean squared difference
between the simulation and performance throughout the contact period, the linear
velocity terms were calculated as a percentage of actual resultant takeoff velocity,
the trunk orientation was simply the difference in orientation in degrees at takeoff
and the angular velocity difference was calculated as the percentage difference at
takeoff. The cost function was a root mean square of these nine component scores,
as shown by Equation 5.2.1.
𝐶𝑂𝑆𝑇 = √𝛥𝑦𝑖
2+𝛥�̇�𝑖2+𝛥�̇�𝑖
2+𝛥𝑦2+𝛥𝑧2+𝛥�̇�2+𝛥�̇�2+𝛥𝜃2+𝛥�̇�2
9
Where: 𝛥𝑦𝑖 = difference in initial horizontal COM position [cm]
𝛥𝑦�̇� = difference in initial horizontal COM velocity [ms-1
]
𝛥𝑧�̇� = difference in initial vertical COM velocity [ms-1
]
𝛥𝑦 = root mean squared difference in horizontal toe position [cm]
𝛥𝑧 = root mean squared difference in vertical toe position [cm]
𝛥�̇� = horizontal takeoff velocity difference [%]
𝛥�̇� = vertical takeoff velocity difference [%]
𝛥𝜃 = takeoff orientation difference [°]
𝛥�̇� = takeoff angular velocity difference [%]
When the seven trials were matched using a common set of parameter values, the
overall score was calculated as the mean of the cost function values across the
seven trials.
5.2.3 Results
When the seven trials were matched the simulated annealing algorithm was able
to optimise the parameter values so that the average score across the seven trials
was 3.3% difference. The individual simulations matched the recorded
performances with scores of 2.2% (S), 4.1% (F1), 4.2% (F2), 3.6% (F3), 2.9% (B1),
2.5% (B2) and 3.5% (B3). Table 5.2.2 shows the common set of parameters
determined by the combined matching procedure. Table 5.2.3 shows the
69
adjustments made to the initial horizontal, vertical, and angular velocities for the
individual simulations.
Table 5.2.2 Common parameter values determined by the combined matching procedure.
Parameter
MT (kg) 18.615
Timp (s) 0.165
Kz1 (N.m-2
) 7531.30
Kz2 (N.m-1
)
-374.138
Ky (N.m-1
) 58418.45
Dz (N.s.m-1
)
0.485
Dy (N.s.m-1
) 7773.38
∇ -1.060
MT – effective mass of trampoline, Timp – impact duration,
Kz1, Kz2 – vertical stiffnesses, Ky – horizontal stiffness, Dz
– vertical damping, Dy – horizontal damping, ∇ - natural
slope.
Table 5.2.3 Adjustments made to the initial horizontal, vertical, and angular velocities for the
individual simulations.
Adjustments to Initial conditions
�̇� �̇� �̇�
(ms-1
) (ms-1
) (°.s-1
)
S1 0.357 0.272 -0.019
F1 0.577 0.200 0.008
F2 0.775 0.621 -0.022
F3 0.679 0.816 -0.031
B1 0.598 0.476 0.051
B2 0.406 0.459 0.049
B3 0.697 0.475 0.020
�̇� = horizontal velocity, �̇� = vertical velocity, �̇� = angular velocity
70
The mean score of 3.3% across the seven selected trials shows that the simulation
model was capable of replicating the motion of a trampolinist both throughout the
contact phase and at takeoff. Table 5.2.5 shows the component scores achieved by
the combined matching procedure for each of the seven skills and the mean value
of each component score.
Table 5.2.4 Cost function component scores of angle-driven simulations.
Component Score Score
𝛥𝑦𝑖 𝛥𝑦�̇� 𝛥𝑧�̇� 𝛥𝑦 𝛥𝑧 𝛥�̇� 𝛥�̇� 𝛥𝜃 𝛥�̇�
(cm) (m/s) (m/s) (cm) (cm) (%) (%) (°) (%) (%)
S1 2.71 1.51 2.84 2.18 2.20 1.81 1.29 3.55 0.05 2.2
F1 0.78 2.01 1.80 2.19 4.37 5.83 4.66 5.20 6.19 4.1
F2 1.75 1.84 2.50 2.93 2.40 8.11 1.44 2.55 7.60 4.2
F3 1.93 2.28 5.16 1.13 2.25 4.53 6.51 1.32 3.28 3.6
B1 1.44 1.48 0.31 1.57 5.75 5.39 1.12 0.71 1.99 2.9
B2 1.49 1.52 0.03 1.58 4.68 4.52 0.85 0.24 2.45 2.5
B3 1.94 1.56 2.62 1.85 4.77 6.63 3.13 1.14 4.22 3.5
Mean 1.72 1.74 2.18 1.92 3.77 5.26 2.71 2.10 3.68 3.3
𝛥𝑦𝑖 – difference in initial horizontal COM position, 𝛥𝑦�̇� – difference in initial horizontal COM
velocity, 𝛥𝑧�̇� – difference in initial vertical COM velocity, 𝛥𝑦 – RMS difference in horizontal toe
position, 𝛥𝑧 – RMS difference in vertical toe position, 𝛥�̇� – horizontal takeoff velocity difference,
𝛥�̇� – horizontal takeoff velocity difference, 𝛥𝜃 – takeoff orientation difference , 𝛥�̇� – takeoff
angular velocity difference.
The initial motion characteristics of the centre of mass were matched closely
across all of the skills, because the optimisation procedure allowed the initial
horizontal, vertical and angular velocities to vary. The largest component score
was found in the difference in horizontal velocity at takeoff (5.26%).
71
5.3 Evaluation of the Angle-Driven Model
In order to ensure that the set of parameter values obtained through the combined
matching procedure could be applied generally to all trampoline impacts and were
not just specific to the seven selected skills, the angle-driven model was
evaluated. The evaluation was conducted by selecting two new skills, a 2¾ tucked
front somersault (F4) and a tucked double back somersault (B4), and simulating
these skills using the parameter values determined by the combined matching
procedure. The initial conditions of the two skills selected for the evaluation can
be found in Table 5.3.1.
Table 5.3.1 Initial conditions of two skills used in the evaluation procedure.
Skill
Initial condition F4 B4
Horizontal toe position (m) 0.143 0.492
Vertical toe position (m) 0.087 0.104
Horizontal toe velocity (ms-1
) 0.465 0.393
Vertical toe velocity (ms-1
) -7.28 -7.74
Orientation angle (°) 75.9 74.9
Angular velocity (°s-1
) 6.95 -14.8
The scores for the two skills used for evaluating the angle-driven model were
calculated using Equation 5.2.1. The simulations were found to match very
closely with the recorded performances as the simulations returned scores of 3.6%
and 2.3% for F4 and B4 respectively.
Figures 5.3.1 and 5.3.2 show visual representations for comparison of the
trampoline contact phase of the recorded and simulated performances of F4 and
B4.
72
Figure 5.3.1 Comparison of F4 trampoline contact phase between recorded performance (above)
and simulated performance (below).
73
Figure 5.3.2 Comparison of B4 trampoline contact phase between recorded performance (above)
and simulated performance (below).
5.4 Chapter Summary
The angle-driven simulation model was employed to determine the visco-elastic
properties of the trampoline suspension system. The combined matching
procedure produced a common set of parameters for the group of seven skills. The
common set of parameters was then applied to simulations of two other skills
which also matched closely to the recorded performances. The next chapter details
the application of a torque-driven simulation model of trampolining.
74
Chapter 6: The Torque-Driven Model
6.1 Chapter Overview
In this chapter the details of the torque-driven simulation model are described.
The methods used to scale the strength of torque generators and to evaluate the
model in the fixed strength protocol are described. The results from the strength
scaling and fixed strength protocols are detailed and discussed.
6.2 Structure of the Torque-Driven Model
The torque-driven simulation model is of identical construction to the angle-
driven model, with seven segments representing the body of the trampolinist in
two dimensions. The movements of the ankle, knee, hip and shoulder joints are
driven by four pairs of torque generators, whilst the metatarsal-phalangeal and
elbow joints are angle-driven. The reaction force exerted by the trampoline on the
feet acts in the same manner as on the angle-driven model. At the four torque-
driven joints a pair of torque generators act, one to exert an extensor torque and
one to exert a flexor torque, as shown by Figure 6.2.1.
75
Figure 6.2.1 A seven-segment torque-driven model of a trampolinist.
The inputs for the torque-driven model are the initial conditions determined from
kinematic data, the parameters governing the foot-suspension system interactions
that were determined using the angle-driven simulation model (as described in
Chapter 4), body segmental inertia parameters, strength parameters, and the
activation profiles for each of the individual torque generators that was used to
drive the simulation. The initial conditions, determined just prior contact, were the
horizontal and vertical position and velocities of the toe, initial joint angles and
angular velocities, as well as the orientation angle and angular velocity of the
trunk.
Throughout a simulation the movements of the torque-driven joints of the
simulation model were driven by the joint torques, which were governed by
activation profiles of each torque generator. The movements of the angle-driven
joints were governed by joint angle time histories at the elbow, whilst the
metatarsal-phalangeal joint was governed by a function of the depression of the
HE HF
KF KE
AP AD
SE
SF
76
trampoline suspension system. The output of the model included the joint angle
and orientation angle time histories, time histories of the horizontal and vertical
mass centre velocities, and whole–body angular momentum at takeoff.
6.3 Torque Generators
Extensor and flexor torque generators at the ankle, knee, hip, and shoulder joints
were used to drive the simulations. The eight torque generator actions were ankle
plantar flexion (AP), ankle dorsi flexion (AD), knee extension (KE), knee flexion
(KF), hip extension (HE), hip flexion (HF), shoulder extension (SE), and shoulder
flexion (SF). Each of the ankle, knee, and hip torque generators was modelled as a
muscle-tendon complex consisting of a contractile component (CON) and series
elastic component (SEC), whilst the shoulder torque generators were modelled
with only a contractile component.
Figure 6.3.1 depicts the muscle tendon complex at a joint, where the joint angle
(θ) is comprised of two angles representing the contractile component (θcon) and
series elastic component (θsec).
Figure 6.3.1 The muscle tendon complex consisting of a contractile component and series elastic
component for both (a) a joint extensor and (b) a joint flexor.
θ θ
θcon
θcon
θsec
θsec
a) b)
77
(6.3.1)
(6.3.2)
(6.3.3)
The geometric relationship between the joint angle (θ), contractile component
angle (θcon) and series elastic component angle (θsec) are defined by the
relationships:
Flexors:
𝜃 = 𝜃𝑐𝑜𝑛 + 𝜃𝑠𝑒𝑐
Extensors:
𝜃 = 2𝜋 − (𝜃𝑐𝑜𝑛 + 𝜃𝑠𝑒𝑐)
The muscle-tendon complex consists of an elastic component in series with a
contractile component, therefore the torque across the series elastic component,
Tsec, must be equal to the torque across the contractile component, Tcon. The torque
produced by the contractile component is dependent on the contractile component
angle and angular velocity, and the strength of the muscle across the joint; this
relationship is detailed in Section 6.3.1. The torque produced by the series elastic
component, Tsec, is dependent on the stiffness, ksec, and θsec, such that:
𝑇𝑠𝑒𝑐 = 𝑘𝑠𝑒𝑐 ∙ 𝜃𝑠𝑒𝑐
6.3.1 Torque – Angle – Angular Velocity Relationship
The maximum voluntary torque that is capable of being produced about a joint has
been found to be dependent on the angle and angular velocity of that joint Yeadon
et al. (2006). All torque generators were modelled as to represent monoarticular
muscles and were not affected by movements at adjacent joints as biarticular
torque generators have been shown to only be necessary when the knee is flexed
by 40° or more (Lewis et al., 2012). As the limit of the range of motion observed
during the recorded performances was 45° and the majority of the movements
were outside this range, the advantages of biarticular torques over monoarticular
torques were assumed to be negligible.
6.3.1.1 Torque – Velocity Relationship
The torque – velocity relationship is described by a four-parameter function
governing the maximum voluntary torque produced over the range of contractile
78
(6.3.4)
(6.3.5)
component velocities, and a three-parameter function defining the differential
activation of muscles during concentric and eccentric actions.
The four parameters that describe the maximum voluntary torque produced over
the range of contractile component velocities are: the maximum torque Tmax in the
eccentric phase, the isometric torque T0, the angular velocity ωmax at which the
curve reaches zero torque, and ωc defined by the vertical asymptote ω=-ωc of the
Hill hyperbola (Hill, 1938).
During concentric muscle action the torque – velocity curve was given by a
rotational equivalent of the classic Hill hyperbola:
𝑇 =𝐶
(𝜔𝑐+ 𝜔)− 𝑇𝑐 (if ω ≤ 0),
Where 𝑇𝑐 = 𝑇0𝜔𝑐
𝜔𝑚𝑎𝑥 , 𝐶 = 𝑇𝑐(𝜔𝑚𝑎𝑥 + 𝜔𝑐).
During eccentric muscle action the relationship between T and ωwas represented
by the rectangular hyperbola:
𝑇 =𝐸
(𝜔𝐸+ 𝜔)− 𝑇𝑚𝑎𝑥 (if ω ≥ 0),
Where 𝜔𝐸 = (𝑇𝑚𝑎𝑥 − 𝑇0)
𝑘𝑇0
𝜔𝑚𝑎𝑥𝜔𝑐
(𝜔𝑚𝑎𝑥 + 𝜔𝑐) , 𝐸 = −(𝑇𝑚𝑎𝑥 − 𝑇0)𝜔𝑒, and k is the ratio of
the slopes of the eccentric and concentric functions at ω = 0, the value of which
Huxley (1957) predicted, 4.3, was used.
The torque – velocity curve describing both concentric and eccentric phases is
shown by Figure 6.3.2.
79
(6.3.6)
Figure 6.3.2 The four-parameter maximum torque function comprising branches of two
rectangular hyperbolas with asymptotes T = -Tc and ω = -ωc; and T = Tmax and ω
= ωe (Yeadon et al., 2006).
6.3.1.2 Differential Activation
During eccentric voluntary contractions, neural inhibition prevents the muscle
from achieving full activation (Westing et al., 1991). Forrester et al. (2011)
defined the differential activation of muscles during concentric and eccentric
actions using a sigmoid function. Three parameters governed the relationship: the
lowest level of activation in the eccentric phase amin, the angular velocity ω1 at the
point of inflection of the function, and a parameter ωr that described the rate at
which the activation increases from amin to amax (≈ 10 ωr). The differential
activation was defined by Equation 6.3.6.
𝑎 = 𝑎𝑚𝑖𝑛 +(𝑎𝑚𝑎𝑥− 𝑎𝑚𝑖𝑛)
[1 +exp(−(𝜔− 𝜔1)
𝜔𝑟)]
Where the maximum activation level amax was assumed to be equal to 1.0.
ω = -ωc
T = Tmax
T0
ωmax
ω = ωe
T = -Tc
80
(6.3.7)
Figure 6.3.3 The three parameter differential activation function in which the activation a rises
from amin to amax with a point of inflection at ω = ω1 (Forrester et al., 2011).
The four-parameter function (Equations 6.3.4 and 6.3.5) and three-parameter
functions (Equation 6.3.6) were multiplied together to give a seven-parameter
function defining the maximum voluntary torque as a function of contractile
component angular velocity.
6.3.1.3 Torque – Angle Relationship
In addition to varying with the angular velocity of the contractile component, the
maximum voluntary torque produced by a joint also varies with the length of the
contractile component, in the situation of a torque generator described by the
angle θcon. The torque – angle relationship was represented by a bell curve
(Edman & Regianni, 1987).
𝑡𝑎 = 𝑒−(𝜃−𝜃𝑜𝑝𝑡)2 (2𝑘2
2)⁄
Where θopt = angle at which maximum torque can be produced
k2 = width of the curve
The two-parameter function defining the torque – angle relationship was
multiplied by the seven-parameter function defining the torque – angular velocity
relationship, giving a nine-parameter function representing the torque – angle –
angular velocity relationship.
81
6.3.2 Strength Parameters
Each torque generator requires nine parameters to govern the torque – angle –
angular velocity relationship of the contractile component and an additional
parameter defining the stiffness of the series elastic component. These parameter
values were obtained from previous studies before being scaled within a matching
procedure employed to evaluate the torque driven model.
Allen (2009) and Jackson (2010) collected joint torque data, from a triple jumper
and gymnast respectively, using an isovelocity dynamometer. The data was
subsequently fitted with a surface representing the nine-parameter torque – angle
– angular velocity relationship to determine the parameter values. Series elastic
component stiffness parameters were estimated based on the properties of the
major muscle groups contributing to the motion (Allen, 2009; Jackson, 2010). The
torque parameters for the hip, knee, and ankle joints were taken from the values
measured from the gymnast by Jackson (2010) and the parameters for the ankle
were taken from Allen’s measurements of the triple jumper (2009). The joint
torque parameters are given in Table 6.3.1.
Table 6.3.1 Torque generator strength parameter values.
Parameter
Tmax T0 ωmax ωc amin ω1 ωr k2 θopt ksec
(Nm) (Nm) (s-1) (s-1) (s-1) (s-1) (rad) (rad) (Nm.rad-1)
AP 351 206 30.80 15.38 0.88 1.38 0.40 0.37 4.22 641
AD 107 64 26.00 3.90 0.99 -1.57 0.44 0.44 2.13 195
KE 421 301 38.44 2.88 0.81 -0.10 0.10 0.51 4.08 805
KF 147 105 33.26 5.43 0.80 -0.13 0.15 1.11 2.11 173
HE 315 225 18.06 2.20 0.78 0.74 0.14 1.06 4.21 1004
HF 235 168 18.36 4.59 0.80 -0.10 0.13 1.22 2.40 306
SF 130 93 38.14 9.52 0.88 -0.13 0.08 1.29 3.35 1574
SE 202 144 32.50 8.13 0.75 -0.06 0.38 1.83 3.96 1988
82
(6.3.8)
(6.3.9)
The effects of using joint torque parameters that were not specific to the subject
used within the kinematic data collection procedure were minimised by scaling
the isometric torque values to those required during the matching procedure. The
process used to accomplish this is detailed in Section 6.3.4
As the contractile and elastic components are in series, the torque expressed by
the contractile component Tcon is equal to the series elastic component torque Tsec.
The initial value (t = 0) of Tcon is calculated assuming the contractile component
angular velocity (�̇�con) is equal to the joint angular velocity (�̇�). Using Equations
6.3.1-3, an iteration calculated the value of 𝜃con for which Tcon and Tsec were
equal. After this initial time step, 𝜃con was updated by integration, assuming
constant velocity:
𝜃con = 𝜃con + �̇�con𝑑𝑡
The series elastic component angle 𝜃sec was then determined using Equations
6.3.1 and 6.3.2, before Tsec was calculated using Equation 6.3.3, and Tcon was
equated to Tsec in order to determine �̇�con.
6.3.3 Muscle Activation Profiles
The nine–parameter, torque – angle – angular velocity function calculates the
maximum voluntary torque that can be produced about a joint at a given
contractile component angle and angular velocity. In order to calculate the torque
applied at the joint T(t) it is necessary to multiply the maximum voluntary torque
Tvol by a muscle activation level A(t):
𝑇(𝑡) = 𝐴(𝑡) ∙ 𝑇𝑣𝑜𝑙 , where 0 ≤ 𝐴(𝑡) ≤ 1
A quintic function with zero velocity and acceleration at the end points was used
to ramp up and down the muscle activation levels (Yeadon and Hiley, 2000):
83
(6.3.10)
𝐴(𝑡) = 𝑎𝑖 + (𝑎𝑓 − 𝑎𝑖) (𝑡−𝑡𝑖
𝑡𝑓− 𝑡𝑖)3
(6 (𝑡−𝑡𝑖
𝑡𝑓− 𝑡𝑖)2
− 15 (𝑡−𝑡𝑖
𝑡𝑓− 𝑡𝑖) + 10)
where A(t) is the activation level at time t, ai is the initial activation level at time ti
and af is the final activation level at time tf.
The muscle activation profiles allowed two separate rampings and were defined
by seven parameters; three activation levels a0, a1 and a2, the start time of the two
ramps ts1 and ts2, and two ramping durations tr1 and tr2.
Figure 6.3.4 Example of a muscle activation profile.
Table 6.3.2 Seven parameters defining muscle activation profiles.
Parameter Definition
a0 Pre-impact, initial activation level
a1 Maximal/minimal activation level
a2 Final activation level
ts Start time of the first ramp
tr1 Duration of the first ramp
i Time interval between first and second ramps
tr2 Duration of the second ramp
0
1Activation
Time t ts
a1
a0 a2
i tr1 tr2
84
6.3.4 Strength Parameter Scaling Factors
The strength of torque-driven simulation model is dependent on the strength
parameters governing the torque generators used to drive the motion. In order to
successfully evaluate a simulation model it must be able to reproduce realistic
movements, so the strength parameters governing the performance of the torque
generators must be able to closely represent the capabilities of the subject.
In order to accurately represent the strength of the subject using strength
parameters measured from different individuals, the isometric torque parameters
were scaled so that the model was able to achieve the torques required to match
the recorded performances (King et al., 2009). During the matching procedure
used to evaluate the torque-driven model, the isometric torque parameters T0 were
multiplied by a scaling factor x to give a subject-specific, scaled isometric torque
sT0. A common value for x was used for both the extensors and flexors at each
joint, so that the relative strength of the extensors and flexors about a joint was
maintained. Whilst performing this process, it was necessary to force the extensor
torque generators to use maximal activation and reduce the level of co-contraction
of the flexor to a minimum, so that the matching procedure was not able to use the
scaling factor to compensate for sub-maximal activation levels or unnecessary co-
contraction:
𝑠𝑇0 = 𝑇0 ∙ 𝑥 when 𝑎𝑚𝑎𝑥 = 1.0
During the model evaluation three trials were matched, providing three values of x
for each of the pairs of torque generators about each joint. From the three values
the highest scaling factor was selected as this represented the largest value of the
isometric torque parameter required by the simulation model in order for it to be
able to perform the movements recorded during the collection of kinematic data.
After the strength scaling factor for each joint had been determined, this value was
fixed and the model was re-evaluated allowing each torque generator to peak at
sub-maximal activation.
(6.3.11)
85
6.3.5 Metatarsal-Phalangeal Joint
The metatarsal-phalangeal joint is angle-driven and is governed by a function that
relates the joint angle to the depression of the trampoline suspension system. The
movement of the MTP joint during the performances was found to be related to
depression of the trampoline through observation of the recorded performances.
During the depression phase of the contact there it was observed that the
relationship between MTP joint angle and trampoline depression was consistent
(Figure 6.3.5). Whilst during the recoil phase it was observed that the MTP joint
angle was only consistent between skills rotating in the same direction (Figure
6.3.6 and Figure 6.3.7)
Figure 6.3.5 Illustration of the relationship between MTP joint angle and trampoline depression
during the depression phase of the trampoline contact.
Figure 6.3.6 Illustration of the relationship between MTP joint angle and trampoline depression in
forward rotating skills during the recoil phase of the trampoline contact.
0.00
0.50
1.00
1.50
2.00
2.50
3.00
-1.00 -0.50 0.00 0.50
MTP
an
gle
(ra
dia
ns)
Trampoline Depression (m)
0.00
0.50
1.00
1.50
2.00
2.50
3.00
-1.00 -0.50 0.00 0.50
MTP
an
gle
(ra
dia
ns)
Trampoline Depression (m)
86
Figure 6.3.7 Illustration of the relationship between MTP joint angle and trampoline depression in
backward rotating skills during the recoil phase of the trampoline contact.
A quintic polynomial was fit to each of the data sets to describe the MTP joint
angle as a function of trampoline depression. The polynomials describing the
depression phase for all skills and recoil phases for forward skills achieved R2
values of 0.456 and 0.365 respectively. The polynomials were used to drive the
MTP joint angle throughout the simulations whilst the first and second derivatives
were used to calculate the angular velocity and acceleration.
Using two functions to describe the MTP joint angle during one simulation meant
that at the transition between the two functions at the lowest point there was a
small discontinuity in the MTP angle and velocity. This small discontinuity was
avoided by subtracting a quadratic function from the polynomial describing the
recoil phase so that at maximum depression the quadratic function was equal to
the difference between the MTP joint angles given by the two functions and the
gradient equal to the difference between the MTP joint angular velocities.
6.4 Model Evaluation
The torque-driven model was evaluated in order to ensure its validity and
reliability. A successful evaluation proves that a simulation model is capable of
accurately reproducing realistic human movement, and can be achieved by
0.00
0.50
1.00
1.50
2.00
2.50
3.00
-1.00 -0.50 0.00 0.50
MTP
an
gle
(ra
dia
ns)
Trampoline Depression (m)
87
comparing simulations to actual performances. Once a simulation model has been
successfully evaluated, and good agreement between simulated and actual
performance has been shown, the model can be used for further analysis and to
simulate motion. The model evaluation was concurrently conducted along with
the process to determine strength parameter scaling factors detailed in Section
6.3.4.
The model was evaluated in a two-step procedure. The first step, a maximal
activation protocol, was carried out to discover the strength scaling parameters
required to enable the simulation model to perform the skills by forcing the
extensors of the supporting joints to be fully activated during the simulations
whilst allowing the strength scaling factors to vary. The second step, a fixed
strength protocol, used a fixed isometric strength for each joint and allowed
submaximal activation of each joint during the simulations.
Three trials were selected to be used in the matching procedure. These trials
covered a broad range of forward rotational requirements. The selected skills were
a single straight forward somersault F1, a piked 1¾ forward somersault F2, and a
piked triffus F3; the same trials as used in the matching of the angle-driven model.
6.4.1 Model Input
The inputs of the torque-driven model were the initial conditions of the
trampolinist immediately prior to first contact with the trampoline, body
segmental inertias, the visco-elastic parameters governing the foot-suspension
system interactions, and the strength parameters governing the each torque
generator. The initial conditions prior to contact were the horizontal and vertical
position and velocities of the toe, orientation angle and angular velocity, and joint
angles and angular velocities determined from the kinematic data. The torques
produced by the torque generators were doubled in order to represent both the left
and right limbs acting symmetrically.
88
6.4.2 Model Variables
Fifty-five parameters were varied within the initial matching procedure; forty-
seven activation parameters, four strength scaling factors, and four parameters to
vary initial conditions.
There were 15 parameters governing the actions of each pair of torque generators,
seven activation parameters each and a strength scaling factor. However because
the strength were being scaled, as described in Section 6.3.4, each of the ankle,
knee, and hip extensors were forced to reach maximal activation, reducing the
number of parameters by three. The flexor torque generators of the ankle and hip
were also forced to reach zero activation to prevent co-contraction artificially
inflating the strength required at the joints, removing a further two parameters. It
was also possible to eliminate four initial activation parameters, a0, as it was
observed from the data that each of the torque-driven joints had a low angular
velocity at the moment of impact with the trampoline. It was reasoned that the
initial extensor and flexor torques must be similar. On this basis the initial
activation level of the extensor was related to the initial activation level of the
flexor in an inverse ratio of their respective isometric strengths, so that the co-
contraction at the joint caused zero net torque at impact.
Based on the torques generated about each joint in the angle-driven simulations
(Appendix 4) the activation profiles of the ankle, knee, and hip torque generators
were constrained so that the extensor torques ramped down and then up, and the
flexor torques ramped up and then down. The activation profiles maintained
flexibility though as the activation level was permitted to remain constant
throughout the simulation or only ramp in one direction if the start times of each
ramp were delayed.
The 47 muscle activation parameters varied within the matching procedures were
limited to values based on information found in the literature.
Yeadon et al. (2010) demonstrated that prior to a landing the muscles of the legs
are activated. The onset of pre-landing activation has been shown to begin up to
139 ms prior to landing dependent on the vertical velocity at impact (Arampatzis
89
et al., 2003), thus the it was decided to set the initial starting time ts to between -
150 and 0 ms, where touchdown occurred at 0 ms.
All joints were given pre-landing activation levels, a0, greater than zero to ensure
co-contraction at all joints and prevent rapid joint accelerations at the beginning of
the simulation. The level of pre-landing activation in a drop jump has been found
to be about 50% of maximal activation (Horita et al., 2002) or higher (Arampatzis
et al., 2003), therefore the pre-landing activation level was set to between 0.2 and
0.6.
Freund and Budingen (1978) showed that the required time to ramp from zero to
maximal activation in voluntary contractions was 70 ms, however it has been
shown that in jumping and landing activities ramp times of between 100 and 200
ms have been required to reach maximal activation (Arampatzis et al., 2003;
Duncan & McDonough, 2000; Jacobs et al., 1996). The lower limit for ramping
durations tr1 and tr2 therefore was set to 100 ms and whilst there is no theoretical
upper limit for the ramp time it was set to 0.55 s for tr1 and 0.3 s for tr2 as by those
times takeoff will have occurred.
The time interval between the two rampings, i, was allowed to vary between 0.0 to
0.15 s. If the second ramp started after takeoff then no second ramp would take
place.
Each of the torque generators were allowed to vary the isometric torque
parameter, by multiplying it by a scaling factor, in order to match the maximum
torques used by the trampolinist during the recorded performances. The scaling
factors were allowed to vary between 0.35 and 2.0 based on the torque values
achieved during the angle-driven simulations.
Four additional parameters were allowed to vary the initial horizontal and vertical
velocity of the toe, the initial orientation angle of the trunk, and the initial angular
velocity of the shoulder.
90
Table 6.4.1 Upper and lower bounds for muscle activation parameters.
Parameter LB UB
a0 0.005 0.6
a1 0.0 1.0
a2 0.0 1.0
ts (s) -0.02 0.02
tr1 (s) 0.1 0.55
i (s) 0.1 0.4
tr2 (s) 0.1 0.3
Extensors
a1 0.0 a0
a2 a1 1.0
Flexors
a1 a0 1.0
a2 0.0 a1
After the initial matching procedure the strength scaling parameters that were
determined to be required for the simulation model to perform the skills were used
in a second matching procedure that allowed submaximal activation of the torque
generators whilst the isometric strength of each torque generator was now fixed.
During the fixed strength matching procedure sixty parameters were varied; seven
activation parameters for each torque generator, and four parameters allowing the
initial conditions to vary. There were no longer the constraints of the extensors
maximal activation, the ankle and hip flexors reaching minimal activation, and the
initial activation levels of each pair of torque generators were no longer
constrained to produce zero net torque.
6.4.3 Cost Function
The torque-driven model was evaluated using a cost function developed to
objectively assess the agreement between the torque-driven simulation and the
recorded performance. The cost function was a root mean square of component
91
(6.4.1)
(6.4.2)
scores designed to measure the discrepancies in three different aspects of the
simulation. The three aspects covered the difference in joint and orientation angles
throughout the simulation SRMS, the difference in joint and orientation angles at the
end of the simulation SABS, and the differences in the movement outcomes at the
end of the simulation SMOV.
𝐶𝑂𝑆𝑇 = √𝑆𝑅𝑀𝑆2 + 𝑆𝐴𝐵𝑆
2 + 𝑆𝑀𝑂𝑉2 + 𝑃2
Where: 𝑆𝑅𝑀𝑆 = component score for orientation and configuration angles
during contact
𝑆𝐴𝐵𝑆 = component score for orientation and configuration angles
at takeoff
𝑆𝑀𝑂𝑉 = component score for movement outcomes
P = joint angle penalties (°)
The component score for orientation and configuration angles during contact SRMS
was the mean square RMS difference between the simulation and recorded
performance, measured in degrees, of the trunk orientation angle θt and the four
joint angles: ankle θa, knee θk, hip θh, and shoulder θs.
𝑆𝑅𝑀𝑆 = √𝜃𝑡𝑅𝑀𝑆
2 + 𝜃𝑎𝑅𝑀𝑆2 + 𝜃𝑘𝑅𝑀𝑆
2 + 𝜃ℎ𝑅𝑀𝑆2 + 𝜃𝑠𝑅𝑀𝑆
2
5
Where: 𝜃𝑡𝑅𝑀𝑆 = RMS difference in trunk orientation (°)
𝜃𝑎𝑅𝑀𝑆 = RMS difference in ankle angle (°)
𝜃𝑘𝑅𝑀𝑆 = RMS difference in knee angle (°)
𝜃ℎ𝑅𝑀𝑆 = RMS difference in hip angle (°)
𝜃𝑠𝑅𝑀𝑆 = RMS difference in shoulder angle (°)
The component score for orientation and configuration angles at takeoff SABS was
calculated as the mean square absolute difference between the simulation and
recorded performance of the orientation and four joint angles measured in
degrees.
92
(6.4.3)
(6.4.4)
𝑆𝐴𝐵𝑆 = √𝜃𝑡𝐴𝐵𝑆
2 + 𝜃𝑎𝐴𝐵𝑆2 + 𝜃𝑘𝐴𝐵𝑆
2 + 𝜃ℎ𝐴𝐵𝑆2 + 𝜃𝑠𝐴𝐵𝑆
2
5
Where: 𝜃𝑡𝐴𝐵𝑆 = RMS difference in trunk orientation (°)
𝜃𝑎𝐴𝐵𝑆 = RMS difference in ankle angle (°)
𝜃𝑘𝐴𝐵𝑆 = RMS difference in knee angle (°)
𝜃ℎ𝐴𝐵𝑆 = RMS difference in hip angle (°)
𝜃𝑠𝐴𝐵𝑆 = RMS difference in shoulder angle (°)
The component score for the movement outcomes was the mean square of
individual component scores measuring the discrepancies in four areas; horizontal
and vertical linear momentum, angular momentum and the duration of the contact
phase. The horizontal linear momentum and vertical linear momentum scores S1
and S2 were calculated as the difference in horizontal and vertical takeoff
velocities as a percentage of recorded resultant velocity at takeoff. The angular
momentum score S3 was calculated as the percentage difference in angular
momentum at takeoff between the recorded and simulated performance. The
contact duration score S4 was calculated as the percentage difference between the
simulated contact duration and the recorded contact duration as a percentage of
the recorded contact duration.
𝑆𝑀𝑂𝑉 = √𝑆12+ 𝑆2
2+ 𝑆32+ 𝑆4
2
4
Where: S1 = percentage difference in horizontal linear momentum (%)
S2 = percentage difference vertical linear momentum (%)
S3 = percentage difference angular momentum (%)
S4 = percentage difference contact duration (%)
The simulations were also penalised if the joint angles of the hip, knee, and ankle
exceeded the range of motion demonstrated by the trampolinist during the data
collection. A penalty was incurred when the simulations exceeded these ranges of
motion, this penalty was equal to the square of the cumulative number of degrees
93
the range of motion was exceeded by across the three joints. The limits of the
ranges of motion for each joint are described in Table 6.4.2.
Table 6.4.2 Limits of the range of motion of the joints, as used for penalties.
Joint Lower Limit Upper Limit
(°) (°)
Ankle 93 152
Knee 133 196
Hip 106 213
6.4.4 Strength Scaling Matching Results
The four trials were matched using the simulated annealing algorithm optimising
the parameter values so that the average difference across the three trials was just
4.4%. The individual simulations matched the recorded performances with scores
of 3.3% (F1), 4.9% (F2), and 5.0% (F3). The mean score of 4.4% across the three
selected trials shows that the simulation model was capable of adequately
replicating the motion of a trampolinist both throughout the contact phase and at
takeoff in forward somersaulting skills. Table 6.4.3 shows the component scores
achieved by the combined matching procedure for each of the four skills and the
mean value of each component score.
94
Table 6.4.3 Cost function component scores resulting from the strength scaling matching protocol.
Component Score Score
SRMS SABS S1 S2 S3 S4 P
(°) (°) (%) (%) (%) (%) (°) (%)
F1 2.8 1.5 0.2 1.7 0.0 0.3 – 3.3
F2 4.0 1.0 2.4 3.2 0.3 3.0 – 4.9
F3 4.8 0.9 2.0 1.2 1.2 0.3 – 5.0
Mean 3.9 1.2 1.5 2.0 0.5 1.2 0.0 4.4
S1 – average RMS difference in orientation and configuration angles during contact (°),
S2 - average difference in orientation and configuration angles at takeoff (°), S3 -
percentage difference in horizontal linear momentum (%), S4 - percentage difference
vertical linear momentum (%), S5 - percentage difference angular momentum (%), S6 -
percentage difference contact duration (%), P - joint angle penalties
The torque-driven model was able to closely match the performance of the
trampolinist in all three forward rotating skills. The angles at takeoff (1.2°),
horizontal linear momentum (1.5%), vertical linear momentum of the trampolinist
(2.0%), and angular momentum at takeoff (0.5%) and the duration of contact
(1.2%) were all closely matched by the simulation model. None of the simulations
received penalties for exceeding the observed ranges of motion.
Figure 6.4.1 and Figure 6.4.2 show visual representations of the trampoline
contact phase during the recorded performances and the matched torque-driven
simulations for a single straight front somersault F1 and a piked triffus F3
respectively.
95
Figure 6.4.1 Comparison of F1 trampoline contact phase performance (above) and torque-driven
simulation (below).
96
Figure 6.4.2 Comparison of F3 trampoline contact phase performance (above) and torque-driven
simulation (below).
6.4.4.1 Joint Angles
The difference between the joint angles of the recorded and simulated
performance was weighted heavily in the evaluation of the simulation model; one
component of the cost function quantified the difference between the simulation
and performance throughout the contact phase and another quantified the
difference at the moment of takeoff. The simulation model was able to reasonably
reproduce the joint movements throughout the contact phase and closely matched
the joint angles at takeoff. Figure 6.4.3 compares the joint angles of the ankle,
knee, hip, and shoulder during the skills F1 and F3.
97
It can be seen that generally the model was able to closely match the joint angles
throughout the contact, although the ankle shows greater plantar flexion during
the depression phase of F1 and the shoulder angle during the depression phase of
F3 is less flexed. It can also be seen that upon reaching maximum depression at
the midpoint of the contact phase and during the recoil phase all the joint angles
follow the recorded performances closely.
98
Figure 6.4.3 Comparison of the joint angles during matched simulations (solid line) and
performance (dashed line) of F1 (left) and F3 (right).
80
120
160
0.000 0.175 0.350
An
gle
(°)
Time (s)
Ankle
80
120
160
0.000 0.175 0.350
An
gle
(°)
Time (s)
Ankle
120
160
200
0.000 0.175 0.350
An
gle
(°)
Time (s)
Knee
120
160
200
0.000 0.175 0.350A
ngl
e (
°)
Time (s)
Knee
100
140
180
0.000 0.175 0.350
An
gle
(°)
Time (s)
Hip
100
140
180
0.000 0.175 0.350
An
gle
(°)
Time (s)
Hip
110
150
190
0.000 0.175 0.350
An
gle
(°)
Time (s)
Shoulder
105
145
185
0.000 0.175 0.350
An
gle
(°)
Time (s)
Shoulder
99
6.4.4.2 Joint Torque Activation Profiles
The activation profiles for the six torque generators for the skills F1 and F3 are
shown in Figure 6.4.4, and the parameter values describing the activation profiles
can be found in Appendix 5. The activation time histories show that the ankle
plantar flexors reached maximal activation after both the hip and the knee
extensors in both skills, and the hip extensors ramped the activation level down
earlier than both the knee and ankle. All of the matching simulations used both the
available ramps, up and down, for both the flexor and extensor at the ankle, hip,
and shoulder, however the knee flexor did not ramp the activation back up in
either F2 or F3.
The hip activation profiles are very different to each other during the recoil phase.
In F1 co-contraction is maintained throughout, however during the recoil phase of
F3 the hip extensors switch off completely and immediately before takeoff the hip
flexors ramp up quickly, in order to promote hip flexion and initiating the forward
motion of the centre of mass in order to produce somersault rotation. This effect
can be seen in Figure 6.4.3 the hip angle in F3 the hip angle quickly decreases
before takeoff, requiring a lower extensor torque.
100
Figure 6.4.4 Activation time histories of the extensors (solid line) and flexors (dashed line) of F1
(left) and F3 (right).
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee
0.0
0.5
1.0
0.000 0.175 0.350A
ctiv
atio
n
Time (s)
Knee
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder
101
6.4.4.3 Joint Torques
The net joint torque time histories obtained from the torque driven simulations for
the skills F1 and F3 are detailed in Figure 6.4.5, with negative torques representing
extension and positive torques representing flexion.
There are similarities between the shapes of the different joint torque; each
supporting joint steadily increased the extensor torque to a maximum point at
around the time of maximum depression of the trampoline before decreasing
during the recoil phase. The joint torque profiles at each joint have similar shapes
for both of the skills but there were some differences in the magnitude and timing
of the torques. The magnitude of the peak extensor torque is greater at each
supporting joint in the triffus with differences in the shape of the torque profiles in
the recoil phase displaying how the joint angles are changed for produce angular
momentum as histories shown in Figure 6.4.3.
During the recoil phase of the triffus the hip extensor torque quickly decreases due
to the sharp deactivation of the hip extensors, whilst also causing a step in the
knee torque as it decreases. The change in the hip torque and the resulting
decrease in the hip angle, also leads to the shoulder flexor torque being
maintained throughout the recoil phase, as a greater torque is required to keep the
arms overhead and produce the necessary arm action when the upper body is
leaning forward.
102
Figure 6.4.5 Comparison of the joint torques during matched torque-driven simulations of F1 (left)
and F3 (right).
-600
-200
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Ankle
-600
-200
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Ankle
-200
0
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Knee
-200
0
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Knee
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Hip
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Hip
-200
0
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Shoulder
-200
0
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Shoulder
103
6.4.4.4 Strength Scaling Factors
The matching procedure varied the strength scaling factors resulting in three
different values, one from each skill, for the scaling factor at each joint. The
resulting strength scaling factors are shown in Table 6.4.4. None of the scaling
factors pushed against the limits of the bounds set during the matching procedure
demonstrating that the strengths of each pair of torque generators was not too
closely constrained to the values measured from the studies of Allen (2009) and
Jackson (2010).
Table 6.4.4 Strength scaling factors.
Strength Scaling Factors
ankle knee hip shoulder
F1 1.04 0.75 1.26 0.47
F2 1.55 0.44 1.34 0.57
F3 1.77 0.93 0.99 0.69
Maximum 1.77 0.93 1.34 0.69
The maximum value of each of the strength scaling factors found across the four
matching simulations was assumed to represent the maximal functional strength
of the trampolinist about each joint whilst trampolining. These values were then
used in a second matching procedure, the results of which are detailed in Section
6.4.5. Three of the maximum values, for the ankle, knee, and shoulder, were
obtained from the matching simulation of the triffus skill; the maximum scaling
factor for the hip came from the 1¾ somersault.
The largest strength scaling factor required in the matching process was 1.77 for
the ankle in F3. This is a large increase over the measured isometric strength of a
triple jumper’s ankle, however given the difficulties in collecting maximal torque
data from the ankle joint this was not deemed to be unreasonable.
104
6.4.5 Fixed Strength Matching Results
The same three trials that were matched previously were subsequently matched
using the simulated annealing algorithm to optimise only the parameter values
governing the activation profiles. This procedure resulted in an average difference
across the 3 trials of just 4.4%. The individual simulations matched the recorded
performances with scores of 3.2% (F1), 4.8% (F2), and 5.3% (F3). The mean score
of 4.4% across the three selected trials shows that the simulation model was
capable of adequately replicating the motion of a trampolinist both throughout the
contact phase and at takeoff. Table 6.4.5 shows the component scores achieved by
the combined matching procedure for each of the three skills and the mean value
of each component score.
Table 6.4.5 Cost function component scores resulting from the fixed strength matching.
Component Score Score
SRMS SABS S1 S2 S3 S4 P
(°) (°) (%) (%) (%) (%) (°) (%)
F1 2.7 1.5 0.2 1.3 0.2 0.0 – 3.2
F2 4.1 1.2 1.4 2.8 0.1 3.0 – 4.8
F3 4.9 1.2 1.7 2.2 1.2 0.6 – 5.3
Mean 3.9 1.3 1.1 2.1 0.5 1.2 0.0 4.4
S1 – average RMS difference in orientation and configuration angles during contact (°),
S2 - average difference in orientation and configuration angles at takeoff (°), S3 -
percentage difference in horizontal linear momentum (%), S4 - percentage difference
vertical linear momentum (%), S5 - percentage difference angular momentum (%), S6 -
percentage difference contact duration (%), P - joint angle penalties
In the fixed strength protocol the torque-driven simulation model was capable of
matching the performance of the trampolinist to the same level as during the
strength scaling matching protocol (4.4% v 4.4%). The fixed strength protocol
achieved the same results largely by reproducing similar average values for each
of the component scores with only some small variation in each of the trials. In
105
both protocols the component scores were evenly balanced and showed a strong
match across the different aspects of the simulation.
Figure 6.4.6 and Figure 6.4.7 show visual representations of the trampoline
contact phase during the recorded performances and the matched torque-driven
simulations for skills F1 and F3 respectively.
Figure 6.4.6 Comparison of F1 trampoline contact phase performance (above) and fixed strength
torque-driven simulation (below).
106
Figure 6.4.7 Comparison of F3 trampoline contact phase performance (above) and fixed strength
torque-driven simulation (below).
6.4.5.1 Joint Angles
Figure 6.4.8 compares the joint angles of the ankle, knee and hip throughout the
performance and fixed strength matched simulations of skills F1 and F3. The fixed
strength matching procedure was able to match the joint angles throughout the
whole duration of the simulations, and showed very similar patterns to the joint
angles in the matched simulations in the protocol. The shoulder angle during the
simulation of F3 followed the same pattern, reducing the initial shoulder velocity
and not matching well in the first half of the contact phase in favour of matching
closely throughout the recoil phase and at takeoff.
107
Figure 6.4.8 Comparison of the joint angles during fixed strength matched simulations (solid line)
and performance (dashed line) of F1 (left) and F3 (right).
80
120
160
0.000 0.175 0.350
An
gle
(°)
Time (s)
Ankle
80
120
160
0.000 0.175 0.350
An
gle
(°)
Time (s)
Ankle
120
160
200
0.000 0.175 0.350
An
gle
(°)
Time (s)
Knee
120
160
200
0.000 0.175 0.350
An
gle
(°)
Time (s)
Knee
100
140
180
0.000 0.175 0.350
An
gle
(°)
Time (s)
Hip
100
140
180
0.000 0.175 0.350
An
gle
(°)
Time (s)
Hip
110
150
190
0.000 0.175 0.350
An
gle
(°)
Time (s)
Shoulder
110
150
190
0.000 0.175 0.350
An
gle
(°)
Time (s)
Shoulder
108
6.4.5.2 Joint Torque Activation Profiles
The activation profiles for the six torque generators for the fixed strength
matching simulations of skills F1 and F3 are shown in Figure 6.4.9, and the
parameter values describing the activation profiles can be found in Appendix 6.
Whilst generally the shapes of the activation profiles are similar to the activation
profiles resulting from the strength scaling matching protocol, due to the strength
of each torque generator having been scaled and the removal of other constraints
the absolute activation levels and relative activation levels of extensors and
flexors have changed. The most pronounced changes were made in the activation
profiles of the ankle joint; in F1 the plantar flexor ramped down activation
minimally in the recoil phase, and in F3 the dorsi flexors only slowly ramped up
their activation through the contact phase instead of ramping off and on.
However, the largest difference in the activation profiles between the strength
scaling matching and the fixed strength matching protocols is the disappearance of
the increase in hip flexor activation immediately prior to takeoff in the triffus, F3,
instead opting to slowly ramp up in a similar profile to the dorsi flexors.
109
Figure 6.4.9 Matched activation time histories of the extensors (solid line) and flexors (dashed
line) of F1 (left) and F3 (right) from the fixed strength protocol.
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee
0.0
0.5
1.0
0.000 0.175 0.350A
ctiv
atio
n
Time (s)
Knee
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder
110
6.4.5.3 Joint Torques
The net joint torque time histories obtained from the fixed strength torque-driven
simulations for the skills F1 and F3 are compared to the joint torque time histories
from the strength scaled matched simulations in Figure 6.4.10, with negative
torques representing extension and positive torques representing flexion.
Each of the joint torque profiles have very similar shapes from both matching
protocols with the only notable difference being the hip torque profile in the
triffus. The fixed strength matching simulation utilised a greater peak hip extensor
torque in the contact phase than the strength scaling matching simulation whilst
the overall shape of the joint torque profiles remained consistent with the strength
scaling protocol.
111
Figure 6.4.10 Comparison of the joint torques during fixed strength matched torque-driven
simulations (solid line) and strength scaling matched simulations (dashed line) of
F1 (left) and F3 (right).
-600
-200
200
0.000 0.175 0.350To
rqu
e (
Nm
)
Time (s)
Ankle
-600
-200
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Ankle
-200
0
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Knee
-200
0
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Knee
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Hip
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Hip
-200
0
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Shoulder
-200
0
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Shoulder
112
6.4.6 Discussion
The evaluation of the torque-driven model has shown that the simulation is
capable of closely representing the performance of forward rotating somersault
skills on the trampoline. After scaling the strength parameters governing the
torque generators to the performance of the subject, the simulation model matched
the performance of the trampolinist well with a mean difference between the
simulations and performances of 4.4%.
6.4.6.1 Joint Angles
The evaluation of the torque-driven model was able to match the joint angles well
throughout the contact phase with the only exception being found at the shoulder
during the depression phase of the triffus. The shapes of the joint angle time
histories were reproduced with only small deviations from the recorded angles in
the majority of cases. The matching simulations seemed to match the joint angles
more closely during the recoil phase than the depression phase of contact, this
most likely arose due to the large weighting of differences in both joint angles and
movement outcomes at takeoff compared to throughout the whole duration of the
movement.
In both of the matching protocols it can be seen that the root mean square
difference in joint angles throughout the contact phase increases as the angular
rotation requirement of the somersaulting skill increases. It can also be seen that
the absolute difference in joint angles at takeoff decreases as the angular rotation
requirement of the somersaulting skill increases. These results also in agreement
with the idea that the composition of the cost function caused the optimisation
algorithm to more closely match the movements of the simulation model to the
complex performances at the moment of takeoff, rather than throughout the whole
duration of the movements.
6.4.6.2 Movement Outcomes at Takeoff
Both matching protocols were able to recreate the movement outcomes of
horizontal and vertical linear momentum, and angular momentum at takeoff, as
well as the duration of the contact phase, to a high degree, with no individual
113
score exceeding 3.3% difference in any of these criteria. Angular momentum was
matched particularly closely in both protocols with no difference greater than
1.2%, whilst the simulation model found it most difficult to match vertical linear
momentum at takeoff, but even this achieved an average difference of almost 2%.
6.4.6.3 Summary
The torque-driven simulation model was demonstrated to be capable of
realistically simulating forward somersaulting trampolining performances. Whilst
the simulation model did not employ any strength parameters that were specific to
the subject, the model was able to accurately represent the strength capabilities of
the subject through the scaling of strength parameters taken from the literature.
The model simulated the performances of the subject and was able to replicate key
performance outcomes of the performances including the linear and angular
momenta at takeoff, and so the model is considered to be suitable to be applied to
optimise performance of forward somersaulting skills.
6.5 Chapter Summary
This chapter described the torque-driven simulation model, the method used to
determine strength scaling factors to be used in conjunction with the simulation
model, and the method used to evaluate the torque-driven simulation model. The
model was able to simulate the performances of the subject well with a mean
difference of 4.4% observed over three different skills, and so is considered
suitable to be employed to answer the research questions. The following chapter
will apply the simulation model of trampolining to answer the research questions.
114
Chapter 7: Optimisation and Applications
7.1 Chapter Overview
The present chapter describes the methods employed to apply the torque-driven
simulation model of trampolining to answer the research questions presented at
the beginning of this thesis. The results are then described and discussed.
7.2 Application to Research Questions
The research questions presented at the beginning of this thesis concerned the optimal
techniques used by a trampolinist during the contact phase with the trampoline.
Specifically:
For specific skills with a fixed rotational requirement what is the optimal takeoff
technique to produce the required angular momentum with maximum peak height
and minimum travel?
and
What is the optimal takeoff strategy to produce maximal somersault rotation
potential in forward somersaults?
Within the present study the trampolinist performed a range of forward rotating skills
with different rotation requirements, ranging up to a piked triffus. Could the subject
perform a straight triffus or possibly a greater number of somersaults by using a different
technique?
In order to perform skills with a larger amount of rotation the trampolinist must be able to
either remain airborne for a longer period of time or takeoff with a greater amount of
angular momentum, or a combination of both. A technique which produces more angular
momentum is likely to reduce the height reached and the time spent airborne. An optimal
technique for producing rotation will enable the trampolinist to takeoff with a high
amount of both angular and vertical linear momenta.
115
(7.3.1)
(7.3.2)
(7.3.3)
7.3 Calculation of Kinematic Variables
7.3.1 Jump Height
The height of the jump hj, was defined as the maximum height of the
trampolinist’s centre of mass above the level of the trampoline suspension system
during the flight phase following the trampoline contact phase. Jump height
comprises two constituent components; the height of the centre of mass at takeoff
h1, and the height gained by the centre of mass during flight h2. Jump height was
calculated as the sum of h1 and h2.
ℎ2 = 𝑣𝑧
2
−2𝑔
ℎ𝑗 = ℎ1 + ℎ2
Where: g = acceleration due to gravity (-9.81 ms-2
)
hj = jump height (m)
h1 = height of CM at takeoff (m)
h2 = height gained by the centre of mass during flight (m)
vz = trampolinist’s CM vertical takeoff velocity (ms-1
)
7.3.2 Flight Time
The flight time t, was defined as the time between the last instant of contact
between the feet of the trampolinist and the trampoline suspension system until
the trampolinist’s centre of mass was 0.936 m above the trampoline, i.e. the
average height of the centre of mass at the moment of touchdown across all
recorded trials. Using constant acceleration equations t was defined as:
𝑡 = 𝑣𝑧+ √(𝑣𝑧
2−2𝑔(0.936 − ℎ1))
𝑔
Where: t = flight time (s)
g = acceleration due to gravity (-9.81 ms-2
)
vz = trampolinist’s CM vertical takeoff velocity (ms-1
)
h1 = height of CM at takeoff (m)
116
(7.3.4)
(7.3.5)
7.3.3 Rotation Potential
Rotation potential is a measure of the trampolinist’s potential to rotate in the flight
phase following takeoff, accounting for the trampolinist’s orientation and angular
momentum at takeoff, as well as the duration of the flight phase following takeoff.
Rotation potential is a normalised product of angular momentum and flight time,
and is expressed as the number of somersaults the trampolinist would be capable
of completing in a straight position.
The somersault angle at takeoff θi, was based on the orientation of the trunk and
thigh at takeoff, and was weighted to represent the relative masses of the upper
and lower body:
𝜃𝑖 = 𝑚𝑢𝜃𝑢 + 𝑚𝑙𝜃𝑙
Where: ml = lower body mass ratio
mu = upper body mass ratio
θi = initial somersault angle (°)
θl = angle of the thigh from vertical (°)
θu = angle of the trunk from vertical (°)
The amount of rotation the trampolinist is capable of during flight θf, was
calculated as the product of angular velocity ω, and flight time t.
𝜃𝑓 = 𝜔𝑡
Where: θf = flight rotation angle (°)
ω = angular velocity (ms-1
)
t = flight time (s)
Angular velocity was calculated using the angular momentum at takeoff H, and
the moment of inertia of the trampolinist in a straight position Is. The moment of
inertia of the trampolinist in a straight position (arms adducted) was calculated to
be 10.6 kg.m2 using the inertia model of Yeadon (1990b.).
117
(7.3.6)
(7.3.7)
𝜔 = 𝐻
𝐼𝑠
Where: ω = angular velocity (ms-1
)
H = angular momentum at takeoff (kg.m2.s
-1)
Is = moment of inertia when straight (kg.m2)
The rotational potential θ, was calculated as the sum of the initial somersault angle
θi, and the flight rotation angle θf, converted to give the number of straight
somersaults that could be achieved:
𝜃 = 𝜃𝑖+ 𝜃𝑓
360
Where: 𝜃 = rotation potential
θi = initial somersault angle (°)
θf = flight rotation angle (°)
7.4 Optimisation Method
7.4.1 Maximum Height Method
The optimisation procedure maximised the height reached during the flight phase
for the three skills employed in the evaluation of the model; a straight single
forward somersault F1, a piked 1¾ forward somersault F2, and a piked triffus F3.
The 56 joint torque activation parameters of the torque-driven model were
optimised to determine the optimal technique to produce jump height during the
flight phase. The strength scaling factors determined during the matching
procedure were maintained, whilst the initial conditions (excluding vertical
velocity), and elbow joint angle time histories for the simulations were taken from
the recorded data. The initial vertical velocity was normalised to -7.5 ms-1
but the
initial horizontal velocity of the trampolinist was also allowed to vary so that the
optimisation procedure could use this as a mechanism to reduce horizontal
118
(7.4.1)
velocity at takeoff. The simulated annealing algorithm varied these 57 parameters
to maximise an objective score function that was designed to maximise height
whilst minimising the difference between the simulated rotation potential and that
required to complete the specified skill; effectively to produce the highest possible
jump height for a given rotation potential.
During the optimisation the joint angle penalties described in Section 6.4.3
remained in place to ensure that the resulting simulated techniques were realistic
to the subject. Simulations also incurred penalties for every millimetre of
horizontal travel during the subsequent flight phase over a distance of 1.075 m;
half the length of the jumping zone.
𝑆𝐶𝑂𝑅𝐸 = 100. ℎ𝑗 − ∆𝜃2 − 𝑃𝑎2 − 𝑃𝑡
2
Where: hj = jump height (m)
∆𝜃 = rotation potential difference (°)
Pa = joint angle penalties (°)
Pt = horizontal travel penalties (mm)
7.4.2 Maximum Rotation Method
The optimisation procedure maximised the rotation potential achieved for the
flight phase following the contact under two different conditions dependent on
horizontal travel; one optimisation was permitted to travel the full length of the
trampoline’s jump zone, 2.15 m, whilst the second optimisation was only allowed
to travel half this distance, 1.075 m.
The optimisation procedure for maximum rotation varied the 56 parameters
governing the activation profiles of the 8 torque generators as in the optimisation
for height. The initial conditions were based on the movements of the piked triffus
F3, and a final parameter varied the initial horizontal velocity whilst the other
initial conditions were not allowed to vary. The elbow joint angle time history of
the simulation was also taken from the F3 trial that was used in the matching
procedure. The simulated annealing algorithm varied the 57 parameters in order to
maximise an objective score function that quantified the rotation potential of the
flight phase following the simulated contact phase, once again with penalties
119
(7.4.2)
constraining the joint movements to within the observed range of motion and for
exceeding the permitted amount of horizontal travel.
𝑆𝐶𝑂𝑅𝐸 = 100. 𝜃 − 𝑃𝑎2 − 𝑃𝑡
2
Where: 𝜃 = rotation potential (rev)
Pa = joint angle penalties (°)
Pt = horizontal travel penalties (mm)
7.5 Optimisation Results
7.5.1 Maximum Jump Height Results
The optimisation procedure was able to increase jump height in each of the skills
by 14%, 9%, and 14% for the single straight forward somersault F1, piked 1¾
forward somersault F2, and piked triffus F3 respectively over the matched
simulations. The simulation model was able to perform a single straight
somersault at a height of 4.16 m, a piked 1¾ somersault at 4.07 m and a piked
triffus at a height of 3.91 m. This was achieved by increases in centre of mass
height at takeoff of 5%, 2%, and 12% respectively and increases in vertical
velocity at takeoff of 8%, 5% and 7% respectively, leading to respective increases
in the time of flight of 9%, 6%, and 8%. The optimal techniques to produce height
in each skill were also capable of matching the rotation potential required to
perform the skills very closely with an average difference of just -0.02% over the
three trials.
120
Table 7.5.1 Differences between matched simulations (M) and the optimal simulation (O) for
maximum jump height in the skills F1, F2, and F3.
F1 F2 F3
M O % Diff M O % Diff M O % Diff
h1 (m) 0.97 1.02 5 0.94 0.96 2 0.81 0.90 12
vz (ms-1
) 7.2 7.8 8 7.4 7.8 5 7.2 7.7 7
t (s) 1.48 1.61 9 1.51 1.60 6 1.44 1.56 8
H (kgm2s
-1) 34.1 33.5 -2 45.5 42.2 -7 70.1 69.1 -1
θ (rev) 0.84 0.84 1 1.07 1.07 0 1.74 1.73 -1
tr (m) 0.35 0.14 1.07
hj (m) 3.64 4.16 14 3.74 4.07 9 3.42 3.91 14
h1 = height of CM at takeoff, vz = trampolinist’s CM vertical takeoff velocity, t = flight time, H =
angular momentum at takeoff, θ= rotation potential measured in straight somersaults, tr = travel,
hj = jump height
The optimised single somersault only resulted in 0.35 m of horizontal travel and
the optimised 1¾ somersault travelled just 0.14 m but the triffus resulted in over
one metre of travel, travelling 1.07 m horizontally. It is possible the triffus could
be performed at a greater height if it was allowed to travel more than half the
length of the jumping zone.
Figures 7.5.1, 7.5.2, 7.5.3 compare the optimal techniques to achieve height to the
technique employed by the matching simulations in the somersaulting trials F1,
F2, and F3 respectively.
121
Figure 7.5.1 Comparison of fixed strength torque-driven simulation of F1 (above) and optimal
technique to produce jump height in a single straight front somersault (below).
122
Figure 7.5.2 Comparison of fixed strength torque-driven simulation of F2 (above) and optimal
technique to produce jump height in a 1¾ piked front somersault (below).
123
Figure 7.5.3 Comparison of fixed strength torque-driven simulation of F3 (above) and optimal
technique to produce jump height in a piked triffus (below).
7.5.1.1 Joint Angles
Figure 7.5.4 compares the joint angle time histories of the ankle, knee, hip, and
shoulder during the optimal solutions for jump height to the fixed strength
matched simulations of the forward somersaulting skills. During the depression
phase the joint angles of the ankle, hip, and shoulder are not much different to the
techniques of the matching simulation whilst the knee angles for the optimal
simulations of the single straight somersault and the piked triffus extend more
slowly than the matched simulations during the second half of the trampoline’s
depression. As the trampoline recoils knee extension occurs later than in the
matching simulations but, in the 1¾ somersault and triffus, extends more quickly
after the delayed onset of the movement. The shoulder also shows differences in
the optimal technique when compared to the matched simulations; the single
somersault maintains a flexed shoulder throughout the recoil phase, the 1¾
124
somersault displays extra extension of the shoulder prior to takeoff, and the triffus
technique shows shoulder extension occurring much slower than during the
matched simulation. During the recoil phase the ankle plantar flexes to a greater
degree than the matched simulations in each of the optimised techniques, the hip
shows some additional extension immediately prior to takeoff in the 1¾
somersault and triffus, whilst the hip extends to a greater degree. Immediately
before takeoff there is an additional plantar flexion of the ankle to provide a last
boost of vertical linear momentum.
125
Figure 7.5.4 Comparison of the joint angle time histories during optimal jump height technique (solid lines) and the fixed strength matched simulation (dashed lines) of F1
(top), F2 (middle), and F3 (bottom).
80
120
160
0.000 0.175 0.350
An
gle
(°)
Time (s)
Ankle
120
160
200
0.000 0.175 0.350
An
gle
(°)
Time (s)
Knee
90
130
170
0.000 0.175 0.350
An
gle
(°)
Time (s)
Hip
105
145
185
0.000 0.175 0.350
An
gle
(°)
Time (s)
Shoulder
80
120
160
0.000 0.175 0.350
An
gle
(°)
Time (s)
Ankle
120
160
200
0.000 0.175 0.350
An
gle
(°)
Time (s)
Knee
90
130
170
0.000 0.175 0.350
An
gle
(°)
Time (s)
Hip
105
145
185
0.000 0.175 0.350
An
gle
(°)
Time (s)
Shoulder
80
120
160
0.000 0.175 0.350
An
gle
(°)
Time (s)
Ankle
120
160
200
0.000 0.175 0.350
An
gle
(°)
Time (s)
Knee
90
130
170
0.000 0.175 0.350
An
gle
(°)
Time (s)
Hip
105
145
185
0.000 0.175 0.350
An
gle
(°)
Time (s)
Shoulder
126
7.5.1.2 Joint Torque Activation Profiles
Figures 7.5.5, 7.5.6, and 7.5.7 compare the joint torque activation profiles utilised
at the ankle, knee, hip, and shoulder during the optimal techniques to produce
jump height and the matching simulations of F1, F2, and F3.
The optimal solution for the single straight somersault shows each of the torque
generator activation profiles follow similar shapes to the activation profiles from
the matching simulation. The knee flexor activation profile is very similar to the
matching simulation whilst the other profiles show combinations of slightly
different activation levels and delayed ramp times. The plantar flexors and knee
extensors ramp up to a greater activation level; whilst in the recoil phase the knee
extensors ramp down later and more slowly. The shoulder flexors meanwhile
show more moderate activation levels, and the shoulder extensors activate more
slowly before takeoff.
The optimal technique for the production of height during a 1¾ piked somersault
was accomplished using activation profiles very similar to the matched simulation
throughout the depression phase, and throughout the whole contact phase the
activation profiles of the knee and shoulder showed only small alterations to those
used by the matching simulations, with only a small difference in the final
activation of the shoulder flexors. The ankle plantar flexors also used a similar
activation profile, but the activation of the dorsi flexors ramped up later and to a
lower level than in the matched solution, whilst the hip extensors ramped down to
an increased level and the hip flexors delayed ramping up to a decreased level.
Optimal height in a piked triffus was achieved by using different activations at all
joints than in the matching solution, with the most notable differences being a
greater level of activation of the hip extensors throughout the contact phase, the
knee flexors decreasing to minimal activation in the recoil phase, and a much
lower amount of plantar flexor activity at takeoff. After the initial ramp down the
shoulder extensors maintained a constant activation through the contact phase
instead of ramping up before takeoff, and the dorsi flexors also employed a
similar activation profile.
127
Figure 7.5.5 Comparison of activation time histories of the optimal solution for jump height (solid
line) and matched simulation (dashed line) of F1.
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle Plantar Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle Dorsi Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee Extensors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip Extensors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder Extensors
128
Figure 7.5.6 Comparison of activation time histories of the optimal solution for jump height (solid
line) and matched simulation (dashed line) of F2.
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle Plantar Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle Dorsi Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee Extensors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip Extensors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder Extensors
129
Figure 7.5.7 Comparison of activation time histories of the optimal solution for jump height (solid
line) and matched simulation (dashed line) of F3.
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle Plantar Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle Dorsi Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee Extensors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip Extensors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder Extensors
130
7.5.1.3 Joint Torques
Figure 7.5.8 compares the joint torque time histories of the ankle, knee, hip, and
shoulder to the optimal solutions for jump height to the fixed strength matching
simulations of the skills F1, F2, and F3.
The differences in the activation profiles of the optimal and matched solutions for
the single straight somersault led to greater extensor torques at the ankle, knee,
and hip in the depression phase, throughout the movement and around the time of
maximal depression respectively. During the recoil phase the net torque at the hip
switched to a flexor torque whereas the matched simulation only used extensor
torques. The shoulder torque showed a smaller extensor torque during the
depression phase.
The optimal solution for the production of height during a 1¾ piked somersault
used similar joint torques at the ankle and knee to the matched solution
throughout the contact phase, whilst the knee employed a significantly greater
extensor torque around the time of maximal trampoline depression. The shoulder
torque during the optimal solution is very similar to the matched solution until the
trampolinist has begun the recoil phase, then the shoulder flexor torque decreases
instead of increasing before takeoff.
The ankle and hip joint torques employed by the simulation model to optimise
jump height in the piked triffus skill were very similar to the matched simulation,
and the shoulder torque followed a similar pattern but did not peak the extensor
torque around maximal depression and maintained an extensor torque through to
takeoff. The major difference in joint torques during the optimal triffus technique
was the magnitude of the hip extensor torques throughout the recoil phase. The
hip extensor torque followed a similar pattern as the matching simulation the
torque was more than twice as large during the recoil phase.
131
Figure 7.5.8 Comparison of the joint torques during optimal jump height technique (solid lines) and the fixed strength matched simulation (dashed lines) of F1 (top), F2
(middle), and F3 (bottom).
-600
-200
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Ankle
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Knee
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Hip
-100
0
100
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Shoulder
-600
-200
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Ankle
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Knee
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Hip
-100
0
100
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Shoulder
-600
-200
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Ankle
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Knee
-400
-100
200
0.000 0.175 0.350To
rqu
e (
Nm
)
Time (s)
Hip
-100
0
100
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Shoulder
132
7.5.1.4 Summary
The optimal solutions achieved jump heights of 4.16 m, 4.07 m and 3.91 m above
the trampoline, decreasing as the rotational requirement increased. This
represented increases of 14%, 9% and 14% over the matched solutions.
The optimal techniques to produce jump height utilised a larger hip and knee
extensor torques around the point of maximal depression and during the recoil
phase than the matched solutions. This was caused by combinations of increased
extensor activation and decreased flexor activation.
During the optimal solutions the both the ankle and hip extensors reach maximal
or near maximal activation (1.000 and 0.997 respectively), possibly indicating that
hip extensor and ankle plantar flexor strength are two of the limiting factors for
the production of jump height in forward rotating somersault.
7.5.2 Maximum Rotation Results
The optimisation of technique to produce rotation potential for the two different
amounts of permissible horizontal travel was able to produce a flight phase during
which 1.91 straight somersaults could be completed whilst only travelling half the
length of the jumping zone 0.5L, and 2.00 straight somersaults whilst travelling
the whole length of the jumping zone 1L. These optimal results represented a 10%
and 15% increase in rotation potential over the matched solution for F3
respectively. The optimal rotation potential was achieved by taking off having
already rotated through 48° and 57°, representing increases of 13% and 33%, and
angular momenta of 83 kg.m2.s
-1 and 90 kg.m
2.s
-1, respective increases of 19%
and 28%. This increased angular momentum was combined with slightly shorter
time in the air to result in increased angles rotated through during flight, as the
flight times achieved by the optimal technique were 1% and 5% shorter than the
matched solution of F3 but during flight the optimal techniques rotated through
angles of 640° and 662° respectively, representing increases of 17% and 21%.
Both optimised techniques used the limits of their permitted horizontal travel.
When allowed to travel half the length of the jumping zone the optimal solution
reached within 4 mm of the limit, whilst when allowed to travel the full length of
133
the jumping zone the optimal solution travelled to within 3 cm of the limit,
however no penalties were incurred by the optimised simulations.
Table 7.5.2 Differences between matched simulations (M) and the optimal simulation (O) for
maximum rotation potential for two limits of travel 0.5L and 1L.
0.5L 1L
M O % Diff M O % Diff
θi (°) 42.7 48.3 13 42.7 56.6 33
vz (ms-1
) 7.2 7.1 -1 7.2 6.8 -5
t (s) 1.44 1.42 -1 1.44 1.37 -5
H (kgm2s
-1) 70.1 83.4 19 70.1 89.5 28
θf (°) 546 640 17 546 662 21
tr (m) 1.07 2.12
θ (rev) 1.74 1.91 10 1.74 2.00 15
θ i = initial somersault angle, vz = trampolinist’s CM vertical takeoff
velocity, t = flight time, H = angular momentum at takeoff, θf = flight
rotation angle, tr = travel, θ = rotation potential measured in straight
somersaults
Figures 7.5.9 and 7.5.10 compare the optimal techniques to achieve maximum
rotation to the fixed strength matching simulation of skill F3.
134
Figure 7.5.9 Comparison of fixed strength torque-driven simulation of F3 (above) and optimal
technique to produce rotation potential with up to 1.075 m of travel (below).
135
Figure 7.5.10 Comparison of fixed strength torque-driven simulation of F3 (above) and optimal
technique to produce rotation potential with up to 2.15 m of travel (below).
7.5.2.1 Joint Angles
The joint angles of the ankle, knee, and hip follow similar patterns to those used
in the matching simulation of skill F3; however there are major differences in the
movements of the shoulder during the recoil phase in both optimised techniques.
Both optimised techniques show significantly less shoulder extension throughout
the recoil phase than the technique used by the trampolinist, this will result in
moving the centre of mass further forwards and increasing the torque created
about the mass centre by the vertical force. During the optimised technique of 1L
the shoulders actually flex further immediately prior to takeoff.
136
In addition to this, optimal solutions also show greater hip extension throughout
the depression phase, with the difference in hip angle peaking around the time of
maximum depression of the trampoline. The greater rigidity of the body during
the depression phase can exploit the elasticity of the trampoline by helping to
depress the trampoline further and storing more elastic energy for the recoil phase.
The knee angle also follows a similar pattern to the technique used by the
trampolinist, but during the recoil phase the knee flexes then extends before
takeoff, so that at takeoff the knee angle is similar but it has a greater angular
velocity of extension at this time. Figure 7.5.11 compares the joint angle time
histories of the optimal technique for the production of rotation to the fixed
strength matching simulation of skill F3.
137
Figure 7.5.11 Comparison of the joint angle time histories of the 0.5L (left) and 1L (right) optimal
techniques to and the fixed strength matching simulation of skill F3.
80
120
160
0.000 0.175 0.350
An
gle
(°)
Time (s)
Ankle
80
120
160
0.000 0.175 0.350
An
gle
(°)
Time (s)
Ankle
120
160
200
0.000 0.175 0.350
An
gle
(°)
Time (s)
Knee
120
160
200
0.000 0.175 0.350
An
gle
(°)
Time (s)
Knee
90
130
170
0.000 0.175 0.350
An
gle
(°)
Time (s)
Hip
90
130
170
0.000 0.175 0.350
An
gle
(°)
Time (s)
Hip
100
140
180
0.000 0.175 0.350
An
gle
(°)
Time (s)
Shoulder
100
140
180
0.000 0.175 0.350
An
gle
(°)
Time (s)
Shoulder
138
7.5.2.2 Joint Torque Activation Profiles
Figures 7.5.12 and 7.5.13 compare the joint torque activation profiles utilised at
the ankle, knee, hip and shoulder during the optimal techniques to produce
rotation to those used during the matched simulation of F3.
In the 0.5L optimisation there are major differences in the activation profiles of all
the joint torque generators except the dorsi flexors and the shoulder flexors. The
plantar flexor activation profile is very similar throughout the depression phase
but during the recoil phase the activation ramps down to a much lower level. At
the knee, the extensors do not reach the same level of activation during the middle
of the contact phase and the flexors ramp down much earlier and to a minimal
level. The hip extensors utilise a significantly larger peak activation level in the
depression phase, the hip flexors use a slowly increasing activation throughout the
contact phase instead of the minimal activation used in the matching simulation,
and the shoulder extensors employed a constant activation level throughout the
recoil phase instead of ramping up prior to takeoff.
The optimal activation profiles for the production of rotation potential using the
full length of the jump zone, 1L, were different to those of the matched simulation
in similar ways to the 0.5L optimisation but differed from the matched solution for
F3 less. The dorsi flexor activation profile was once again very similar to the
matched solution, but the shoulder flexor showed greater disparity, ramping down
from peak activation earlier, before maximal depression. The plantar flexor still
ramps down to a lower level than the matching simulation, however not as low as
the 0.5L optimal technique. The knee extensor activation profile reached the same
level as in the matching simulation however the initial ramp up was shorter and
the ramping down of the activation level began earlier, around maximal
depression, and took longer, whilst the knee flexors followed the same pattern but
ramped down to a higher activation level during the recoil phase. At the hip, the
extensors again use a greater peak activation level and ramp up to that level and
ramp down earlier but the peak activation level is closer to that used by the
matching solution than the 0.5L optimal solution. The hip flexors and shoulder
extensors follow very similar activation profiles to the 0.5L solution.
139
Figure 7.5.12 Comparison of activation time histories of the 0.5L optimal solution for rotation
potential (solid line) and matched simulation of F3 (dashed line).
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle Plantar Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle Dorsi Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee Extensors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip Extensors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder Extensors
140
Figure 7.5.13 Comparison of activation time histories of the 1L optimal solution for rotation
potential (solid line) and matched simulation of F3 (dashed line).
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle Plantar Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Ankle Dorsi Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee Extensors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Knee Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip Extensors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Hip Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder Flexors
0.0
0.5
1.0
0.000 0.175 0.350
Act
ivat
ion
Time (s)
Shoulder Extensors
141
7.5.2.3 Joint Torques
Figure 7.5.14 compares the joint torque time histories of the optimal techniques to
produce rotation, 0.5L and 1L, to the fixed strength matching simulation of skill
F3. Both the optimal techniques utilise ankle torques very similar to those of the
matching simulation, knee extensor torques that are much greater during the recoil
phase, lower peak hip extensor torques around maximal depression and hip flexor
torques immediately prior to takeoff, and shoulder torques that move from an
extensor torque to a flexor torque during the recoil phase.
Whilst both optimal techniques use larger knee extensor torques and smaller hip
extensor torques than the matched simulation, as well as similarly different
shoulder torques, the 0.5L solution uses greater knee and hip extensor torques than
the 1L solution, whilst the 1L technique achieves a net shoulder flexor torque
earlier in the contact phase.
142
Figure 7.5.14 Comparison of the joint torque time histories of the optimal techniques, 0.5L (left)
and 1L (right) for the production of rotation potential (solid line) and the fixed
strength matched simulation of skill F3 (dashed line).
-600
-200
200
0.000 0.175 0.350To
rqu
e (
Nm
)
Time (s)
Ankle
-600
-200
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Ankle
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Knee
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Knee
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Hip
-400
-100
200
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Hip
-100
0
100
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Shoulder
-100
0
100
0.000 0.175 0.350
Torq
ue
(N
m)
Time (s)
Shoulder
143
7.5.2.4 Summary
The optimal techniques produced a flight phase during which 1.9 and 2.0 straight
somersaults could be completed for 0.5L and 1L respectively, representing
increases of 10% and 15% over the matched solution for F3.
The optimal techniques to produce rotation potential utilised a larger knee
extensor torques during the recoil phase and lesser hip extensor torques around
peak depression than the matched solution, although the hip extensor activations
were greater and the knee extensor activation being lower than the matching
solution. This effect was caused by the relative activations of the antagonists; the
knee flexor activation levels were lower, and the hip flexors were activated to a
greater level.
During the optimal solutions the ankle plantar flexors reach near maximal
activation (0.999 and 0.990), possibly indicating that plantar flexor strength is one
of the limiting factors for the production of forward somersault rotation.
7.6 Discussion
The optimisation of the technique employed by the simulation model of
trampolining to produce maximum jump height and rotation potential showed that
the trampolinist was employing suboptimal techniques during the data collection.
The optimal techniques were able to increase jump height by up to 14% and
rotation potential by up to 15% over the matched simulations. It was also seen that
even though jump height directly effects the rotation potential, the optimal
technique for rotation potential only achieved a jump height 86% as high as the
optimal technique for jump height, underlining the compromise between the
production of angular and linear momenta.
The optimal techniques used near maximal activations, this possibly demonstrates
that the hip and ankle strength are limiting factors for jump height and ankle
strength is a limiting factor for the production of rotation.
144
7.7 Chapter Summary
This chapter described how the simulation model was applied to answer the
research questions. Optimisation demonstrated that there was potential for the
trampolinist to improve performance by up to 15% by employing changes to his
technique. The findings of the present study, the methods used and future
applications of the model will be discussed in Chapter 8.
145
Chapter 8: Discussion and Conclusions
8.1 Chapter Overview
This chapter summarises the main findings of the present study. The methodology
will be discussed, highlighting the limitations of the current method and
suggesting possible improvements for future research. Finally the potential future
applications of the simulation model of the trampolining contact phase developed
during the present study will be addressed.
8.2 Research Questions
Q1. For specific skills with a fixed rotational requirement what is the optimal
takeoff technique to produce the required angular momentum with maximum peak
height and minimum travel?
The results of this study found that as the rotational requirement of the skill
increased, less height was able to be achieved in the flight phase; this was in
agreement with the findings of previous studies (Miller & Munro, 1984; Sanders
& Wilson, 1988). The optimisation of the simulation model was able to produce
increases in jump height of up to 14% for forward rotating skills with specified
rotation requirements, with increases of up to 8% in vertical takeoff velocity,
identified as the primary factor in increasing jump height by Feltner et al. (1999)
and Sander & Wilson (1992).
For the optimisation of forward rotating trampoline skills to produce maximum
jump height, the optimal techniques utilised were very similar to one another
during the depression phase. The optimised techniques typically showed increased
hip flexion at takeoff and a later and quicker extension of the knee during the
recoil phase as the rotational requirement increased. As well as these
characteristics the optimal techniques for jump height also used greater plantar
flexion during the recoil phase and a less flexed shoulder angle but with slower
rates of extension as the rotational requirement increased. These characteristics of
146
the techniques could cause the angular momentum required for the skills to be
produced later in the contact phase, so that during the early portion of the contact
phase maximum elastic potential energy can be stored in the trampoline bed, and
then in the recoil phase the elastic energy can be used to create maximum vertical
linear momentum before the trampolinist flexes, moving out of the line of action
of the vertical force to create the required rotation.
The differences in technique are primarily the result of changes in the activation
profiles of the ankle, knee, and hip flexors, rather than the extensors, with the
main result being a larger peak hip extensor torque decreasing more quickly
during the recoil phase.
Q2. What is the optimal takeoff strategy to produce maximal somersault rotation
potential in both forward somersaults?
When the takeoff strategy was optimised for maximal somersault rotation, the
simulation model was able to produce up to 15% more rotation potential. The
optimal techniques managed to achieve substantial increases in angular
momentum, up to 28%, with only small reductions in flight time of up to 5%.
The optimal techniques to produce rotation potential both utilised a technique that
involves a large amount of hip flexion and much less shoulder extension during
the recoil phase, and even small amounts of shoulder flexion immediately prior to
takeoff. These movements increase the horizontal displacement of the centre of
mass, increasing the moment arm of the vertical force acting on the feet and
enabling the production of greater quantities of angular momentum. The
techniques also showed greater hip extension at the time of maximal trampoline
depression, and small degrees of knee flexion in the recoil phase followed by a
quick extension of the knee in the moments before takeoff, that could increase the
total vertical force during the recoil phase, and also when the moment arm of the
vertical force is at its largest, utilising the vertical force with the optimum strategy
(Ollerenshaw, 2004; Vaughan, 1980).
These optimal techniques were produced by smaller shoulder flexor and larger
knee extensor torques during the recoil phase, and smaller hip extensor torques
during the depression phase that decreased to a minimal amount earlier in the
147
recoil phase. These torques were the results of not increasing the activation of the
shoulder extensors prior to takeoff, increased activation of the hip extensors
during the depression phase and of the hip flexors during the recoil phase, and
decreased activation of the knee flexors during the recoil phase.
8.3 Discussion
8.3.1 Simulation Model
The simulation model developed in the present study represented a simplification
of trampolining and human motion based on assumptions that may have limited
the capabilities of the model to accurately simulate the performance of a
trampolinist. The trampolinist was modelled by a planar representation assuming
bilateral symmetry. This representation neglects any movements that take place
outside the primary plane of motion, most notably of the arms, and assumes that
the motion is symmetrical when reflected in the plane of motion. The method of
projecting the three dimensional coordinates of the arms onto a plane to calculate
two dimensional joint angles could misrepresent the configuration of the upper
limb and the resulting inaccuracies in the torque time histories may have affected
the orientation and angular momentum of the trampolinist.
The trampolinist was represented by seven rigid segments with the head and trunk
represented by a single rigid segment, assuming that the curvature of the spine
remained constant. This is a major assumption as it was witnessed during the data
collection that prior to takeoff for forward rotating skills flexion of the spine
occurs and hyperextension of the spine occurs before takeoff for backward
rotation skills, such motions could play a major role in the production of angular
momentum. The advantage of modelling the spine as a single rigid segment is that
the torque acting across the length of the spine is much more easily measured than
measuring the torque between multiple spinal segments. The angle-driven
simulations showed hyperextension of the hip and this was also allowed by the
torque-driven model to facilitate the matching process and the production of
angular momentum during the optimisation procedure.
148
The model also did not include any representation of the soft tissue as it had
previously been shown not to be necessary in the simulation of impacts with
compliant surfaces. The inclusion of wobbling masses in the simulation model
would likely have taken a lot of time and resulted in very little effect on the
outputs of the simulations.
8.3.2 Foot-Suspension System Interface
The foot-suspension system interface was modelled by a damped, two
dimensional force-displacement relationship which was non-linear vertically and
linear horizontally. The force-displacement relationship also accounted for the
force required to accelerate the mass of the suspension system. The reaction force
was distributed between the heel, metatarsal-phalangeal joint and toe based on the
relative depressions of the three points compared to the natural slope of the
trampoline around the most depressed point. The trials were selected to be located
centrally on the trampoline, however the impacts were not located in the very
centre of the trampoline and the location has been shown to affect the force-
displacement relationship. The representation of the foot-suspension system did
not take into account the location of impact on the trampoline however the force-
displacement relationship was able to represent the foot-suspension system
interface satisfactorily as the angle-driven model was capable of closely matching
the performance data.
8.3.3 Performance Data
The trampolining performances were recorded using an automatic motion capture
system recording at 300 Hz. The motion of the trampolinist was tracked by opto-
reflective markers placed along the midline of the torso and the joints of the right
limbs. In only recording the movements of the right limbs it was assumed that the
movements of the trampolinist would be symmetrical and the movement of the
right limb would represent the average motion during each trial. An alternate
method could record the motion of both limbs and take an average position of
both of the limbs as an input for a planar simulation model, however due to the
spatial constraints of the available trampolining facility the automatic motion
149
capture system could only be arranged in a configuration suitable to reliably
record the motion of one side of the body.
8.3.4 Anthropometric Data
Anthropometric data was taken from direct measurements of the subject’s body
and body segmental inertia parameters were calculated using the model of Yeadon
(1990b). Inertia parameters calculated using this method have been used in
various rigid body simulation models that have been evaluated and shown to
accurately reproduced human motion King et al., 2009; Wilson et al., 2006;
Yeadon & Hiley, 2000). The method is of limited ability to accurately estimate
the mass of the torso without knowing the volume of air in the lungs however the
effect of this limitation has been shown to be small given the success it has
achieved in simulating a variety of different sporting motions.
Distances between marker locations and joint centres were measured in the frontal
and lateral axes; the distance between each of the markers on the foot and the
floor were also measured. In measuring the joint centre offsets, the precise
location of the joint centre was not clear however reliable estimates of the centre
of rotation were found using manual joint rotation and the assumption that the
joint centre lies along the longitudinal axis of two connecting segments.
8.3.5 Kinematic Data Processing
The kinematic data was processed by transposing the recorded marker locations to
the joint centre locations by adjusting the position data according to the offset
measurements taken from the subject. The elbow, shoulder, hip, knee, and ankle
positions were transposed toward the midline of the body in the frontal plane, and
the metatarsal-phalangeal joint and toe were transposed at right angles to the
vector between that marker and the ankle in the sagittal plane. This process
assumed that the movements of the trampolinist were confined to the sagittal
plane and movements outside of this plane would have affected the kinematic data
used in the matching procedures.
150
8.3.6 Determination of Suspension System Interface Parameters
The parameters governing the force interactions of the foot-suspension system
interface were determined by a combined matching procedure using seven trials of
varying rotation requirements. The matching procedure used a simulated
annealing algorithm to minimise the difference between the angle-driven
simulation model and the recorded performances. The parameter set was then
evaluated by using it to simulate two independent skills, one forward rotating and
one backward rotating, and was found to be applicable to trampoline contacts in
general.
8.3.7 Torque Parameters
The parameters governing the performance of the torque generators were taken
from the previous studies of Allen (2009) and Jackson (2010) who modelled the
motion of triple jumping and gymnastic vaulting respectively. The torque
parameter sets were then scaled to the recorded performances of the trampolinist,
whilst forcing maximal activation to reach 1.0, by multiplying the isometric
torque parameter by a scaling factor which was allowed to vary in the matching
procedure in the evaluation of the torque-driven model using the method
described by (King et al., 2009). The activities of gymnastics and triple jumping
were considered to be reasonably similar to trampolining to assume that the
torque-angle-angular velocity profiles of the individuals to be sufficiently similar.
The method of scaling torque parameters measured from third parties has limited
ability to match the angle and angular velocity dependent aspects of the torque, as
only the isometric torque was scaled to match the maximum torque required by
the recorded performances. Despite this limitation the torque-driven model was
able to simulate the performance of the trampolinist closely and realistically using
the scaled torque parameters and the optimised performance only allowed for a
modest improvement in jump height and reasonable improvement in rotation
potential, suggesting that the torque parameters were realistic assuming the
subject was already using near optimal technique. Ideally accurate torque
parameters could be measured from the subject directly using isovelocity
dynamometry, however measuring torque parameters accurately from a subject is
time consuming. The subject should go through a familiarisation protocol in order
151
to be able to produce maximal effort during the testing procedure, greatly
increasing the time involvement of the subject in the data collection procedure.
Bilateral symmetry was assumed once again, as the torques produced about each
joint were simply doubled to represent the activity of both limbs. The model did
not incorporate any bilateral deficit is the phenomena has been shown to be
primarily caused by increased velocity of shortening (Bobbert et al., 2005) which
is already incorporated by the simulation model.
8.3.8 Evaluation of the Torque-driven Model
The torque-driven model was evaluated by varying the activation parameters and
a scaling factor for each of the torque generators using a simulated annealing
algorithm whilst minimising the difference between the simulations and recorded
performances of forward somersaults. The torque-driven simulations were able to
closely match the recorded performances and so the model was considered to be
suitable for the purpose of simulating forward rotating trampolining skills.
8.3.9 Cost Scores and Penalties
The structure and components of a cost score, that is minimised or maximised
within an optimisation procedure, are the most important factors in the
optimisation process. The constituent components and the weighting of those
components dictate which aspects of the simulation outcomes the optimisation
process focusses on matching to the greatest extent.
The cost scores used in the matching procedures were taken as a root mean square
of multiple component scores describing the difference of many variables between
the simulations and recorded performances. These variables described differences
at the beginning, the end, and throughout simulations, and examined positions,
velocities, orientations and configurations.
In the angle-driven matching procedure the standard deviation of the average
component scores was 0.72 and in the torque driven matching procedure the
standard deviation was 1.91. This shows RMS function was able to distribute the
scores evenly amongst each of the components.
152
The optimisation scores for height and rotation potential heavily weighted the
primary scores of height and rotation potential respectively, and both penalised
the simulations for unrealistic joint angles and unwanted horizontal travel, whilst
the height score also penalised the simulations for differences to the desired
rotation potential. Joint angle penalties were squared, weighting them heavily, so
that unrealistic techniques were not rewarded by the optimisation procedure.
The weighting of the penalty scores applied to each simulation will also heavily
influence the solutions found during the optimisation procedure. If penalties are
weighted too lightly compared to the cost score simulations that violate the
constraints applied might be accepted as long if the overall simulation is still a
good match. During the matching of the torque driven models none of the
matching simulations incurred penalties whilst the optimal techniques were able
to improve on the performance of the subject suggesting the penalties were
weighted correctly.
8.4 Future Research
After a simulation model has been successfully evaluated it can be applied to
answer research questions. The torque-driven model developed in the present
study could be further applied to investigate wider aspects of the trampoline
contact phase and trampoline takeoff techniques including some of the areas
detailed below.
8.4.1 Robustness
Trampolinists are not able to perfectly coordinate movements time after time and
yet they are capable of reliably performing complex skills time after time. It
follows that they are able to use slightly different techniques to accomplish the
same movement outcomes and to make adjustments in order to compensate for
variations in movement characteristics between touchdown and takeoff as well as
during the following flight period. The evaluated simulation model can be used to
investigate how sensitive the simulated motion is to perturbations in the activation
profiles. For example if the trampolinist mistimed the extension of the hips by 10
153
ms would they still be able to perform a piked triffus, or if the optimal technique
used to complete 3 somersaults was perturbed slightly would they still be able to
complete those somersaults or would they lose enough angular momentum to
make attempting the skill dangerous.
8.4.2 Initial Conditions
Maximal depression of an elastic surface has been shown to result in maximal
jump height in both trampolining (Shvartz, 1967) and diving (Cheng & Hubbard,
2004), and increased touchdown velocity has also been associated with increased
jump height (Sanders & Wilson, 1988). Variations in joint configurations at
touchdown will also affect the effectiveness of certain techniques to produce the
desired movement outcomes. Varying the initial horizontal and vertical velocity
of the centre of mass, the initial orientation and angular velocity of the trunk, and
the initial joint configurations would allow the model to investigate optimal
takeoff techniques given variations in initial conditions and examine how those
optimal techniques differ according to the changes in initial conditions.
8.4.3 Subject-Specific Parameters
Throughout time the bodies of athletes can change through training and natural
growth, effecting the strength capabilities and body size of an individual. As the
body grows and changes the optimal takeoff techniques for that athlete will
change to allow them to exploit strength gains or changes in the distribution of
their body mass. Such changes in the optimal techniques can be investigating
through the application of the simulation model with altered strength and body
segmental inertia parameters. The results could have strong implications for
coaching as strength and conditioning programs could be designed to specifically
fit the needs of a given trampolinist, there may also be implications for talent
identification as specific body types may be better suited to trampolining than
others.
8.4.4 Sensitivity to Model Parameters
The parameters governing the foot-suspension system interface in the present
study were determined to be specific to the trampolinist and the trampoline that
154
was used during the data collection. Trampoline beds can have varying degrees of
elasticity and the construction of the suspension system has changed during the
past few decades. The sensitivity of the simulation model to changes in the model
parameters could be assessed by varying the individual parameter values and
investigating the resulting changes in the performance of the model.
8.4.5 Application to Different Trampoline Contacts
Trampolinists perform skills that involve contact between the trampoline and parts
of the body other than the feet. The contact between the trampoline and the
anterior or posterior aspects of the torso could be represented by a simulation
model based upon the model used in the present study to allow the simulation of
skills where a trampolinist lands on their front or back. The primary change to the
model required to simulate these skills would be to alter the distribution of force
between different points along the interacting surfaces.
8.5 Conclusions
The torque-driven simulation model developed and successfully evaluated during
the present study has been applied to optimise trampoline takeoff techniques for
height and rotation potential in both forward rotating and backward rotating
somersaults. The assumptions made during the development of the model and the
limitations of the data collection, parameter determination and optimisation
processes have been discussed, and improvements for future simulation models of
trampolining have been suggested. Future applications of the simulation model
have been considered, including investigations of questions that cannot be
investigated by experimental methods and possible extensions of the present
study.
155
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APPENDICES
169
APPENDIX 1
Informed Consent Form
Appendix 1
170
DATA ACQUISITION FOR THE ANALYSIS OF HUMAN MOVEMENTS
LAY SUMMARY
The study comprises a biomechanical analysis of human movement. This analysis
requires:
Kinematic (how you are moving) during human movements
Subject specific inertia parameters
The data of actual human movements are required to give detailed information about the
current techniques used by humans. The subject specific parameters are required for the
customisation of computer simulation models to individual humans. The simulation
models will then be used to understand and explain techniques currently used, determine
the contributions of different techniques to performance and also optimise performance.
The kinematic, kinetic and EMG data may be obtained in a number of different ways:
Automatic displacement acquisition system. This is similar to being videoed but
reflective markers or LEDs will be taped to you and only their image recorded.
The subject specific parameters may be obtained from:
Anthropometric measurements. Measuring the size of your limbs and body.
Data will be acquired in the biomechanics research facilities in the University or in other
research laboratories. Any data collection session will last no longer than two hours, with
the subject actively involved for only a fraction of the total time:
Actual performance of movements: 15 minutes
Anthropometric measurements: 30 minutes
A medical history questionnaire and full written consent will be required from the parent
(if the subject is under the age of 18) or the subject prior to participation in the study.
DOCUMENTS WHEN SUBJECTS ARE AT LEAST 18 YEARS OF AGE
INFORMATION FOR SUBJECTS
PRE-SELECTION MEDICAL QUESTIONNAIRE
INFORMED CONSENT FORM (SUBJECT)
Appendix 1
171
INFORMATION FOR SUBJECTS
The study in which you have been invited to participate will involve a biomechanical
analysis of human movement. The study will be divided into two parts; firstly, a video
recording will be taken of you performing selected human movements. You will only be
asked to perform movements that you are familiar with and feel comfortable performing
such as those listed:
Trampolining
The second part of the study will involve measurements to determine the lengths, widths
and circumferences of your body segments (e.g. your arms, legs, trunk and head). It may
also be necessary to take additional measurements to estimate your strength
characteristics during various activities (e.g. extending and flexing your knee or hip). The
measurement procedures will be described and demonstrated in advance. It may be
necessary to shave certain areas of your body to attach monitoring equipment using
adhesive tape. The data collected will be used to help increase our understanding of the
mechanics of human movements.
You will perform the data collection in a suitable environment. The risk of injury during
the data collection will be minimal since we will only ask you to perform movements
with which you are familiar and comfortable. It is considered that no increased risks,
discomforts or distresses are likely to result from the data collection of human movements
above those associated with the normal performance of those movements.
The information obtained from the study will be collected and stored in
adherence with the Data Protection Act. Whilst certain personal and
training information will be required, you will be allocated a reference
number to ensure that your identity and personal details will remain
confidential. Video recordings will be stored in the video analysis room to
which access is restricted to members of the biomechanics research team.
The video images will be digitised and only the numerical values will be used
in published work, not the images themselves. On occasion video images may
be required. In such an instance we will seek your written permission to use
such images and you are perfectly free to decline. Video recordings will be
kept for three years after publication of the study. If you agree to take part in
the study, you are free to withdraw from the study at any stage, with or
without having to give any reasons. A contact name and phone number will
be provided to you for use if you have any queries about any part of your
participation in the study.
Appendix 1
172
PRE-SELECTION MEDICAL QUESTIONNAIRE
LOUGHBOROUGH UNIVERSITY
SCHOOL OF SPORT, EXERCISE AND HEALTH SCIENCES
Please read through this questionnaire, BUT DO NOT ANSWER ANY OF THE
QUESTIONS YET. When you have read right through, there may be questions you
would prefer not to answer. Assistance will be provided if you require it to discuss any
questions on this form. In this case please tick the box labelled “I wish to withdraw”
immediately below. Also tick the box labelled “I wish to withdraw” if there is any other
reason for you not to take part. tick appropriate box
I wish to withdraw
I am happy to answer the questionnaire
If you are happy to answer the questions posed below, please proceed. Your answers will
be treated in the strictest confidence.
1. Are you at present recovering from any illness or operation? YES/NO*
2. Are you suffering from or have you suffered from or received medical treatment
for any of the following conditions?
a. Heart or circulation condition YES/NO*
b. High blood pressure YES/NO*
c. Any orthopaedic problems YES/NO*
d. Any muscular problems YES/NO*
e. Asthma or bronchial complaints YES/NO*
3. Are you currently taking any medication that may affect your participation in the
study? YES/NO*
4. Are you recovering from any injury? YES/NO*
5. Are you epileptic? YES/NO*
6. Are you diabetic? YES/NO*
7. Are you allergic to sticking plasters? YES/NO*
8. Do you have any other allergies? If yes, please give details below
YES/NO*
………………………………………………………………………………………….…
………………………………………………………………………………………
9. Are you aware of any other condition or complaint that may be affected by
participation in this study? If so, please state below;
………………………………………………………………………………………….…
………………………………………………………………………………………
* Delete as appropriate
Appendix 1
173
INFORMED CONSENT FORM (SUBJECTS)
PURPOSE
To obtain kinematic data during human movements. To obtain subject specific inertia
parameters.
PROCEDURES
The kinematic data of human movements will be obtained using:
Automatic displacement acquisition system
ACTIVITIES
Possible activities of which only those to be undertaken will be listed
Trampolining
A number of trials will be requested with suitable breaks to minimise fatigue and
boredom.
The subject specific parameters will be obtained from:
Anthropometric measurements (using tape measures and specialist anthropometers)
During the measurements two researchers will be present, at least one of
whom will be of the same sex as you.
QUESTIONS
The researchers will be pleased to answer any questions you may have at any time.
WITHDRAWAL
You are free to withdraw from the study at any stage, with or without having
to give any reasons.
CONFIDENTIALITY
Your identity will remain confidential in any material resulting from this work. Video
recordings will be stored in the video analysis room to which access is restricted to
members of the biomechanics research team. The video images will be digitised and only
the numerical values will be used in published work, not the images themselves. On
occasion video images may be required. In such an instance we will seek your written
permission to use such images and you are perfectly free to decline. Video recordings
will be kept for three years after publication of the study.
I have read the outline of the procedures which are involved in this study, and I
understand what will be required by me. I have had the opportunity to ask for further
information and for clarification of the demands of each of the procedures and understand
what is entailed. I am aware that I have the right to withdraw from the study at any time
with no obligation to give reasons for my decision. As far as I am aware I do not have
any injury or infirmity which would be affected by the procedures outlined.
Name …………………………………………
Signed ………………………………………… (subject) Date ……………
In the presence of:
Name …………………………………………
174
APPENDIX 2
Anthropometric Measurements
Appendix 2a. Anthropometric measurements taken for use in conjunction with the inertia
model of Yeadon (1990b)
Appendix 2b. Marker offset measurements
Appendix 2a
175
Trampolining
Anthropometrics
60.05 kg
168.7 cm
ALL MEASUREMENTS IN MILLIMETRES
TORSO
level → hip umbilicus ribcage nipple shoulder neck → nose ear top
length 0 170 225 428 515 565 0 145 200 300
perimeter 793 700 690 867
351
474 565
width 297 277 254 313 352
depth
156
LEFT
ARM
level → shoulder
Mid-
arm elbow forearm wrist → thumb knuckle nails (3)
length 0
245 284 490 0 60 100 178
perimeter 313 259 249 258 165
227 224 114
width
56
92 82 56
RIGHT
ARM
level → shoulder
Mid-
arm elbow forearm wrist → thumb knuckle nails (3)
length 0
245 293 495 0 60 100 178
perimeter 323 260 247 261 167
232 197 114
width
56
93 82 55
LEFT
LEG
level → hip crotch
Mid-
thigh knee calf ankle → heel arch ball
nails
(3)
ankle →
floor
length 0 105
367 515 775 0 10
145 210 73
perimeter
518 449 367 348 216
313 238 221 150
width
88 63
depth
116
RIGHT
LEG
level → hip crotch
Mid-
thigh knee calf ankle → heel arch ball
nails
(3)
ankle →
floor
length 0 112
375 532 780 0 10
145 210 67
perimeter
524 451 358 350 219
314 246 223 158
width
92 65
depth
117
Appendix 2b
176
Marker Offset Measurements
Axis
x y z z → floor
sternal
75
shoulder 45
elbow 45
wrist 45
hip 100
knee 68
ankle 40
78
ball 0
39 60
toe 0
27 36
177
APPENDIX 3
AutolevTM
4.1 Command Files
Appendix 3a. Autolev
TM commands used to create the Fortran code for the angle-driven
simulation model of a trampolinist
Appendix 3b. AutolevTM
commands used to create the Fortran code for the torque-driven
simulation model of a trampolinist
Appendix 3a
178
% TRAMPANG.AL
% ANGLE-DRIVEN 7-SEGMENT MODEL OF A TRAMPOLINE TAKEOFF
% -TRIANGULAR 2-SEGMENT FOOT
% -FORCE RELATIONSHIP TO BE REPLACED IN FORTRAN CODE
%--------------------------------------------------------------------
% PHYSICAL DECLARATION
NEWTONIAN N
FRAMES T % TRIANGULAR FOOT
BODIES A,B,C,D,E,F,G
POINTS CM,O,P1,P2,P3,P4,P5,P6,P7,P8,P9,P10
AUTOZ ON
%--------------------------------------------------------------------
% MATHEMATICAL DECLARATION
MASS A=MA,B=MB,C=MC,D=MD,E=ME,F=MF,G=MG
INERTIA A,0,0,IA
INERTIA B,0,0,IB
INERTIA C,0,0,IC
INERTIA D,0,0,ID
INERTIA E,0,0,IE
INERTIA F,0,0,IF
INERTIA G,0,0,IG
SPECIFIED QBAL'',QANK'',QKNE'',QHIP'',QSHO'',QELB'',&
TBAL,TANK,TKNE,THIP,TSHO,TELB,& % TORQUE
RX{3},RZ{3} % REACTION FORCES
CONSTANTS THETA,G,& % TRIANGULAR FOOT ANGLE,GRAVITY
L{18} % LENGTHS
VARIABLES Q{3}',U{9}' % DEGREES OF FREEDOM
VARIABLES POAOX,POAOZ,POBOX,POBOZ,POCOX,POCOZ,PODOX,PODOZ,&
POEOX,POEOZ,POP1X,POP1Z,POP10X,POP10Z,&
POP2X,POP2Z,POP3X,POP3Z,POP4X,POP4Z,POP5X,POP5Z,&
POP6X,POP6Z,POP7X,POP7Z,POP8X,POP8Z,POP9X,POP9Z
VARIABLES VOAOX,VOAOZ,VOBOX,VOBOZ,VOCOX,VOCOZ,VODOX,VODOZ,&
VOEOX,VOEOZ,VOP1X,VOP1Z,VOP10X,VOP10Z,&
VOP2X,VOP2Z,VOP3X,VOP3Z,VOP4X,VOP4Z,VOP5X,VOP5Z,&
VOP6X,VOP6Z,VOP7X,VOP7Z,VOP8X,VOP8Z,VOP9X,VOP9Z
VARIABLES AOP1X,AOP1Z,AOP2X,AOP2Z,AOP4X,AOP4Z
VARIABLES POCMX,POCMZ,VOCMX,VOCMZ,AOCMX,AOCMZ,&
KET,KECM,KEA,KEB,KEC,KED,KEE,KEF,KEG,&
PET,PECM,PEA,PEB,PEC,PED,PEE,PEF,PEG,&
M,ANGMOM,HORMOM,VERMOM
ZEE_NOT = [RX1,RX2,RX3,RZ1,RZ2,RZ3,TBAL,TANK,TKNE,THIP,TSHO,TELB]
M = MA + MB + MC + MD + ME + MF + MG
QBAL = T^3 % CALL JANGLES IN .F FILE FOR ANGLE VEL AND
QANK = T^3 % ACC TO OVER-WRITE THE Q VALUES HERE
QKNE = T^3
QHIP = T^3
QSHO = T^3
QELB = T^3
%--------------------------------------------------------------------
% GEOMETRICAL RELATION
SIMPROT(N,E,3,Q3) % TRUNK/HORIZONTAL
SIMPROT(D,E,3,QHIP) % TRUNK/THIGH
SIMPROT(D,C,3,QKNE) % THIGH/SHANK
SIMPROT(B,C,3,QANK) % SHANK/FOOT
SIMPROT(A,B,3,QBAL) % FOOT/TOES
Appendix 3a
179
SIMPROT(B,T,3,THETA) % FIXED TRIANGULAR FOOT
SIMPROT(E,F,3,QSHO) % TRUNK/UPPER ARM
SIMPROT(G,F,3,QELB) % UPPER ARM/FOREARM
%--------------------------------------------------------------------
% POSITION
P_O_P1> = Q1*N1> + Q2*N2>
P_P1_AO> = (L1-L2)*A1>
P_P1_P2> = L1*A1>
P_P2_P3> = -L3*B1>
P_P3_BO> = L4*B1> - L5*B2>
P_P3_P4> = -L6*T1> - L7*T2>
P_P3_CO> = (L8-L9)*C1>
P_P3_P5> = L8*C1>
P_P5_DO> = -(L10-L11)*D1>
P_P5_P6> = -L10*D1>
P_P6_EO> = L13*E1>
P_P6_P7> = L12*E1>
P_P6_P8> = L14*E1>
P_P8_FO> = -L16*F1>
P_P8_P9> = -L15*F1>
P_P9_GO> = L18*G1>
P_P9_P10> = L17*G1>
P_O_AO> = P_O_P1> + P_P1_AO>
P_O_P2> = P_O_P1> + P_P1_P2>
P_O_BO> = P_O_P2> + P_P2_BO>
P_O_P3> = P_O_P2> + P_P2_P3>
P_O_P4> = P_O_P3> + P_P3_P4>
P_O_CO> = P_O_P3> + P_P3_CO>
P_O_P5> = P_O_P3> + P_P3_P5>
P_O_DO> = P_O_P5> + P_P5_DO>
P_O_P6> = P_O_P5> + P_P5_P6>
P_O_EO> = P_O_P6> + P_P6_EO>
P_O_P7> = P_O_P6> + P_P6_P7>
P_O_P8> = P_O_P6> + P_P6_P8>
P_O_FO> = P_O_P8> + P_P8_FO>
P_O_P9> = P_O_P8> + P_P8_P9>
P_O_GO> = P_O_P9> + P_P9_GO>
P_O_P10> = P_O_P9> + P_P9_P10>
P_O_CM> = CM(O)
POP1X = DOT(P_O_P1>,N1>)
POP1Z = DOT(P_O_P1>,N2>)
POP2X = DOT(P_O_P2>,N1>)
POP2Z = DOT(P_O_P2>,N2>)
POP3X = DOT(P_O_P3>,N1>)
POP3Z = DOT(P_O_P3>,N2>)
POP4X = DOT(P_O_P4>,N1>)
POP4Z = DOT(P_O_P4>,N2>)
POP5X = DOT(P_O_P5>,N1>)
POP5Z = DOT(P_O_P5>,N2>)
POP6X = DOT(P_O_P6>,N1>)
POP6Z = DOT(P_O_P6>,N2>)
POP7X = DOT(P_O_P7>,N1>)
POP7Z = DOT(P_O_P7>,N2>)
POP8X = DOT(P_O_P8>,N1>)
POP8Z = DOT(P_O_P8>,N2>)
POP9X = DOT(P_O_P9>,N1>)
POP9Z = DOT(P_O_P9>,N2>)
POP10X = DOT(P_O_P10>,N1>)
POP10Z = DOT(P_O_P10>,N2>)
POAOX = DOT(P_O_AO>,N1>)
POAOZ = DOT(P_O_AO>,N2>)
POBOX = DOT(P_O_BO>,N1>)
POBOZ = DOT(P_O_BO>,N2>)
POCOX = DOT(P_O_CO>,N1>)
Appendix 3a
180
POCOZ = DOT(P_O_CO>,N2>)
PODOX = DOT(P_O_DO>,N1>)
PODOZ = DOT(P_O_DO>,N2>)
POEOX = DOT(P_O_EO>,N1>)
POEOZ = DOT(P_O_EO>,N2>)
POFOX = DOT(P_O_FO>,N1>)
POFOZ = DOT(P_O_FO>,N2>)
POGOX = DOT(P_O_GO>,N1>)
POGOZ = DOT(P_O_GO>,N2>)
POCMX = DOT(P_O_CM>,N1>)
POCMZ = DOT(P_O_CM>,N2>)
%--------------------------------------------------------------------
% KINEMATICAL DIFFERENTIAL EQUATIONS
Q1' = U1
Q2' = U2
Q3' = U3
%--------------------------------------------------------------------
% ANGULAR VELOCITY & ACCELERATIONS
W_E_N> = U3*E3>
W_E_D> = QHIP'*E3> + U4*E3>
W_C_D> = QKNE'*C3> + U5*C3>
W_C_B> = QANK'*C3> + U6*C3>
W_B_A> = QBAL'*B3> + U7*B3>
W_T_B> = 0>
W_F_E> = QSHO'*F3> + U8*F3>
W_F_G> = QELB'*F3> + U9*F3>
ALF_E_N> = U3'*E3>
ALF_E_D> = QHIP''*E3>
ALF_C_D> = QKNE''*C3>
ALF_C_B> = QANK''*C3>
ALF_B_A> = QBAL''*B3>
ALF_T_B> = 0>
ALF_F_E> = QSHO''*F3>
ALF_F_G> = QELB''*F3>
%--------------------------------------------------------------------
% LINEAR VELOCITY
V_O_N> = 0>
V_P1_N> = DT(P_O_P1>,N)
V2PTS(N,A,P1,AO)
V2PTS(N,A,P1,P2)
V2PTS(N,B,P2,BO)
V2PTS(N,B,P2,P3)
V2PTS(N,T,P3,P4)
V2PTS(N,C,P3,CO)
V2PTS(N,C,P3,P5)
V2PTS(N,D,P5,DO)
V2PTS(N,D,P5,P6)
V2PTS(N,E,P6,EO)
V2PTS(N,E,P6,P7)
V2PTS(N,E,P6,P8)
V2PTS(N,F,P8,FO)
V2PTS(N,F,P8,P9)
V2PTS(N,G,P8,GO)
V2PTS(N,G,P8,P10)
V_CM_N> = DT(P_O_CM>,N)
VOP1X = DOT(V_P1_N>,N1>)
VOP1Z = DOT(V_P1_N>,N2>)
VOP2X = DOT(V_P2_N>,N1>)
VOP2Z = DOT(V_P2_N>,N2>)
VOP3X = DOT(V_P3_N>,N1>)
Appendix 3a
181
VOP3Z = DOT(V_P3_N>,N2>)
VOP4X = DOT(V_P4_N>,N1>)
VOP4Z = DOT(V_P4_N>,N2>)
VOP5X = DOT(V_P5_N>,N1>)
VOP5Z = DOT(V_P5_N>,N2>)
VOP6X = DOT(V_P6_N>,N1>)
VOP6Z = DOT(V_P6_N>,N2>)
VOP7X = DOT(V_P7_N>,N1>)
VOP7Z = DOT(V_P7_N>,N2>)
VOP8X = DOT(V_P8_N>,N1>)
VOP8Z = DOT(V_P8_N>,N2>)
VOP9X = DOT(V_P9_N>,N1>)
VOP9Z = DOT(V_P9_N>,N2>)
VOP10X = DOT(V_P10_N>,N1>)
VOP10Z = DOT(V_P10_N>,N2>)
VOAOX = DOT(V_AO_N>,N1>)
VOAOZ = DOT(V_AO_N>,N2>)
VOBOX = DOT(V_BO_N>,N1>)
VOBOZ = DOT(V_BO_N>,N2>)
VOCOX = DOT(V_CO_N>,N1>)
VOCOZ = DOT(V_CO_N>,N2>)
VODOX = DOT(V_DO_N>,N1>)
VODOZ = DOT(V_DO_N>,N2>)
VOEOX = DOT(V_EO_N>,N1>)
VOEOZ = DOT(V_EO_N>,N2>)
VOFOX = DOT(V_FO_N>,N1>)
VOFOZ = DOT(V_FO_N>,N2>)
VOGOX = DOT(V_GO_N>,N1>)
VOGOZ = DOT(V_GO_N>,N2>)
VOCMX = DOT(V_CM_N>,N1>)
VOCMZ = DOT(V_CM_N>,N2>)
%--------------------------------------------------------------------
% LINEAR ACCELERATION
A_O_N> = 0>
A_P1_N> = DT(V_P1_N>,N)
A_P2_N> = DT(V_P2_N>,N)
A_P3_N> = DT(V_P3_N>,N)
A_P4_N> = DT(V_P4_N>,N)
A_P5_N> = DT(V_P5_N>,N)
A_P6_N> = DT(V_P6_N>,N)
A_P7_N> = DT(V_P7_N>,N)
A_P8_N> = DT(V_P8_N>,N)
A_P9_N> = DT(V_P9_N>,N)
A_P10_N> = DT(V_P10_N>,N)
A_AO_N> = DT(V_AO_N>,N)
A_BO_N> = DT(V_BO_N>,N)
A_CO_N> = DT(V_CO_N>,N)
A_DO_N> = DT(V_DO_N>,N)
A_EO_N> = DT(V_EO_N>,N)
A_FO_N> = DT(V_FO_N>,N)
A_GO_N> = DT(V_GO_N>,N)
A_CM_N> = DT(V_CM_N>,N)
AOP1X = DOT(A_P1_N>,N1>)
AOP1Z = DOT(A_P1_N>,N2>)
AOP2X = DOT(A_P2_N>,N1>)
AOP2Z = DOT(A_P2_N>,N2>)
AOP4X = DOT(A_P4_N>,N1>)
AOP4Z = DOT(A_P4_N>,N2>)
AOCMX = DOT(A_CM_N>,N1>)
AOCMZ = DOT(A_CM_N>,N2>)
%--------------------------------------------------------------------
% AUXILIARY CONSTRAIN
AUXILIARY[1] = U4
Appendix 3a
182
AUXILIARY[2] = U5
AUXILIARY[3] = U6
AUXILIARY[4] = U7
AUXILIARY[5] = U8
AUXILIARY[6] = U9
CONSTRAIN (AUXILIARY[U4,U5,U6,U7,U8,U9])
%--------------------------------------------------------------------
% ENERGY
KEA = KE(A)
KEB = KE(B)
KEC = KE(C)
KED = KE(D)
KEE = KE(E)
KEF = KE(F)
KEG = KE(G)
KECM = KE(A,B,C,D,E,F,G)
PEA = -MA*G*POAOZ
PEB = -MB*G*POBOZ
PEC = -MC*G*POCOZ
PED = -MD*G*PODOZ
PEE = -ME*G*POEOZ
PEF = -MF*G*POFOZ
PEG = -MG*G*POGOZ
PECM = -M*G*POCMZ
%--------------------------------------------------------------------
% ANGULAR & LINEAR MOMENTUM
AMOM> = MOMENTUM(ANGULAR,CM)
ANGMOM = DOT(AMOM>,N3>)
LMOM> = MOMENTUM(LINEAR)
HORMOM = DOT(LMOM>,N1>)
VERMOM = DOT(LMOM>,N2>)
%--------------------------------------------------------------------
% FORCES
GRAVITY(G*N2>)
FORCE(P1,RX1*N1> + RZ1*N2>)
FORCE(P2,RX2*N1> + RZ2*N2>)
FORCE(P4,RX3*N1> + RZ3*N2>)
TORQUE(B/A,TBAL*N3>)
TORQUE(C/B,TANK*N3>)
TORQUE(C/D,TKNE*N3>)
TORQUE(E/D,THIP*N3>)
TORQUE(F/E,TSHO*N3>)
TORQUE(F/G,TELB*N3>)
%--------------------------------------------------------------------
% EQUATIONS OF MOTION
ZERO = FR() + FRSTAR()
KANE(TBAL,TANK,TKNE,THIP,TSHO,TELB)
%--------------------------------------------------------------------
% INPUTS
INPUT TINITIAL=0.0,TFINAL=1.0,INTEGSTP=0.001,PRINTINT=100
INPUT ABSERR=1.0E-08,RELERR=1.0E-07
INPUT G=-9.806,THETA=32.543
INPUT MA=0.2762,MB=1.5878,MC=7.7935,MD=14.9787,ME=29.2589,&
Appendix 3a
183
MF=3.2917,MG=3.1395
INPUT Q1=0,Q2=0,Q3=90,U1=0,U2=5,U3=0
INPUT L1=0.0650,L2=0.0283,L3=0.1450,L4=0.0608,L5=0.0171,L6=0.048,&
L7=0.078,L8=0.4065,L9=0.1711,L10=0.371,L11=0.1648,L12=0.8650,&
L13=0.3827,L14=0.5650,L15=0.2450,L16=0.1118,&
L17=0.4255,L18=0.1618
INPUT IA=0.0002,IB=0.0033,IC=0.0996,ID=0.1786,IE=1.5544,&
IF=0.0184,IG=0.0408
%--------------------------------------------------------------------
% OUTPUTS
OUTPUT T,POP1X,POP1Z,POP2X,POP2Z,POP3X,POP3Z,POP4X,POP4Z,&
POP5X,POP5Z,POP6X,POP6Z,POP7X,POP7Z,POP8X,POP8Z,&
POP9X,POP9Z,POP10X,POP10Z,POCMX,POCMZ
OUTPUT T,VOP1X,VOP1Z,VOP2X,VOP2Z,VOP3X,VOP3Z,VOP4X,VOP4Z,&
VOP5X,VOP5Z,VOP6X,VOP6Z,VOP7X,VOP7Z,VOP8X,VOP8Z,&
VOP9X,VOP9Z,VOP10X,VOP10Z
OUTPUT T,POCMX,POCMZ,VOCMX,VOCMZ,AOCMX,AOCMZ
OUTPUT T,Q3,QBAL,QANK,QKNE,QHIP,QSHO,QELB
OUTPUT T,U3,QBAL',QANK',QKNE',QHIP',QSHO',QELB'
OUTPUT T,U3',QBAL'',QANK'',QKNE'',QHIP'',QSHO'',QELB''
OUTPUT T,TBAL,TANK,TKNE,THIP,TSHO,TELB
OUTPUT T,RX1,RZ1,RX2,RZ2,RX3,RZ3
OUTPUT T,HORMOM,VERMOM,ANGMOM
OUTPUT T,KECM,KEA,KEB,KEC,KED,KEE,KEF,KEG
OUTPUT T,PECM,PEA,PEB,PEC,PED,PEE,PEF,PEG
%--------------------------------------------------------------------
% UNITS
UNITS T=S,[M,MA,MB,MC,MD,ME,MF,MG]=KG
UNITS [IA,IB,IC,ID,IE,IF,IG]=KGM^2
UNITS [Q3,QBAL,QANK,QKNE,QHIP,QSHO,QELB,THETA]=RAD
UNITS [U3,QBAL',QANK',QKNE',QHIP',QSHO',QELB']=RAD/S
UNITS [U3',QBAL'',QANK'',QKNE'',QHIP'',QSHO'',QELB'']=RAD/S^2
UNITS [Q1,Q2,L1,L2,L3,L4,L5,L6,L7,L8,L9,L10,L11]=M
UNITS [L12,L13,L14,L15,L16,L17,L18]=M
UNITS [POP1X,POP1Z,POP2X,POP2Z,POP3X,POP3Z,POP4X,POP4Z]=M
UNITS [POP5X,POP5Z,POP6X,POP6Z,POP7X,POP7Z,POP8X,POP8Z]=M
UNITS [POP9X,POP9Z,POP10X,POP10Z,POCMX,POCMZ]=M
UNITS [VOP1X,VOP1Z,VOP2X,VOP2Z,VOP3X,VOP3Z,VOP4X,VOP4Z]=M/S
UNITS [VOP5X,VOP5Z,VOP6X,VOP6Z,VOP7X,VOP7Z,VOP8X,VOP8Z]=M/S
UNITS [VOP9X,VOP9Z,VOP10X,VOP10Z,VOCMX,VOCMZ,U1,U2]=M/S
UNITS [U1',U2',AOCMX,AOCMZ,G]=M/S^2
UNITS [RX1,RX2,RX3,RZ1,RZ2,RZ3]=N
UNITS [TBAL,TANK,TKNE,THIP,TSHO,TELB]=NM
UNITS ANGMOM=KGM^2/S,[HORMOM,VERMOM]=KGM/S
UNITS [KECM,KEA,KEB,KEC,KED,KEE,KEF,KEG]=J
UNITS [PECM,PEA,PEB,PEC,PED,PEE,PEF,PEG]=J
%--------------------------------------------------------------------
SAVE TRAMP_ANG.ALL
CODE DYNAMICS() TRAMP_ANG.F, SUBS
%--------------------------------------------------------------------
% END END END END END END END END END END END END END
%--------------------------------------------------------------------
Appendix 3b
184
% TRAMPTQ.AL
% TORQUE-DRIVEN 7-SEGMENT MODEL OF A TRAMPOLINE TAKEOFF
% -TRIANGULAR 2-SEGMENT FOOT
% -FORCE RELATIONSHIP TO BE REPLACED IN FORTRAN CODE
% -JOINT TORQUES TO BE CALCULATED IN FORTRAN CODE
%--------------------------------------------------------------------
% PHYSICAL DECLARATION
NEWTONIAN N
FRAMES T % TRIANGULAR FOOT
BODIES A,B,C,D,E,F,G
POINTS CM,O,P1,P2,P3,P4,P5,P6,P7,P8,P9,P10
AUTOZ ON
%--------------------------------------------------------------------
% MATHEMATICAL DECLARATION
MASS A=MA,B=MB,C=MC,D=MD,E=ME,F=MF,G=MG
INERTIA A,0,0,IA
INERTIA B,0,0,IB
INERTIA C,0,0,IC
INERTIA D,0,0,ID
INERTIA E,0,0,IE
INERTIA F,0,0,IF
INERTIA G,0,0,IG
SPECIFIED TBAL,TANK,TKNE,THIP,TSHO,TELB,& % TORQUE
RX{3},RZ{3} % REACTION FORCES
CONSTANTS THETA,G,& % TRIANGULAR FOOT ANGLE,GRAVITY
L{18} % LENGTHS
VARIABLES Q{9}' % DEGREES OF FREEDOM
VARIABLES U{9}'
VARIABLES POAOX,POAOZ,POBOX,POBOZ,POCOX,POCOZ,PODOX,PODOZ,&
POEOX,POEOZ,POFOX,POFOZ,POGOX,POGOZ,POP1X,POP1Z,&
POP2X,POP2Z,POP3X,POP3Z,POP4X,POP4Z,POP5X,POP5Z,&
POP6X,POP6Z,POP7X,POP7Z,POP8X,POP8Z,POP9X,POP9Z,&
POP10X,POP10Z
VARIABLES VOAOX,VOAOZ,VOBOX,VOBOZ,VOCOX,VOCOZ,VODOX,VODOZ,&
VOEOX,VOEOZ,VOFOX,VOFOZ,VOGOX,VOGOZ,VOP1X,VOP1Z,&
VOP2X,VOP2Z,VOP3X,VOP3Z,VOP4X,VOP4Z,VOP5X,VOP5Z,&
VOP6X,VOP6Z,VOP7X,VOP7Z,VOP8X,VOP8Z,VOP9X,VOP9Z,&
VOP10X,VOP10Z
VARIABLES POCMX,POCMZ,VOCMX,VOCMZ,AOCMX,AOCMZ,&
KET,KECM,KEA,KEB,KEC,KED,KEE,KEF,KEG,&
PET,PECM,PEA,PEB,PEC,PED,PEE,PEF,PEG,&
M,ANGMOM,HORMOM,VERMOM
ZEE_NOT = [RX1,RX2,RX3,RZ1,RZ2,RZ3]
M = MA + MB + MC + MD + ME + MF + MG
%--------------------------------------------------------------------
% GEOMETRICAL RELATION
SIMPROT(N,E,3,Q3) % TRUNK/HORIZONTAL
SIMPROT(D,E,3,Q7) % TRUNK/THIGH
SIMPROT(D,C,3,Q6) % THIGH/SHANK
SIMPROT(B,C,3,Q5) % SHANK/FOOT
SIMPROT(A,B,3,Q4) % FOOT/TOES
SIMPROT(B,T,3,THETA) % FIXED TRIANGULAR FOOT
SIMPROT(E,F,3,Q8) % TRUNK/UPPER ARM
SIMPROT(G,F,3,Q9) % UPPER ARM/FOREARM
Appendix 3b
185
%--------------------------------------------------------------------
% POSITION
P_O_P1> = Q1*N1> + Q2*N2>
P_P1_AO> = (L1-L2)*A1>
P_P1_P2> = L1*A1>
P_P2_P3> = -L3*B1>
P_P3_BO> = L4*B1> - L5*B2>
P_P3_P4> = -L6*T1> - L7*T2>
P_P3_CO> = (L8-L9)*C1>
P_P3_P5> = L8*C1>
P_P5_DO> = -(L10-L11)*D1>
P_P5_P6> = -L10*D1>
P_P6_EO> = L13*E1>
P_P6_P7> = L12*E1>
P_P6_P8> = L14*E1>
P_P8_FO> = -L16*F1>
P_P8_P9> = -L15*F1>
P_P9_GO> = L18*G1>
P_P9_P10> = L17*G1>
P_O_AO> = P_O_P1> + P_P1_AO>
P_O_P2> = P_O_P1> + P_P1_P2>
P_O_BO> = P_O_P2> + P_P2_BO>
P_O_P3> = P_O_P2> + P_P2_P3>
P_O_P4> = P_O_P3> + P_P3_P4>
P_O_CO> = P_O_P3> + P_P3_CO>
P_O_P5> = P_O_P3> + P_P3_P5>
P_O_DO> = P_O_P5> + P_P5_DO>
P_O_P6> = P_O_P5> + P_P5_P6>
P_O_EO> = P_O_P6> + P_P6_EO>
P_O_P7> = P_O_P6> + P_P6_P7>
P_O_P8> = P_O_P6> + P_P6_P8>
P_O_FO> = P_O_P8> + P_P8_EO>
P_O_P9> = P_O_P8> + P_P8_P9>
P_O_GO> = P_O_P9> + P_P9_GO>
P_O_P10> = P_O_P9> + P_P9_P10>
P_O_CM> = CM(O)
POP1X = DOT(P_O_P1>,N1>)
POP1Z = DOT(P_O_P1>,N2>)
POP2X = DOT(P_O_P2>,N1>)
POP2Z = DOT(P_O_P2>,N2>)
POP3X = DOT(P_O_P3>,N1>)
POP3Z = DOT(P_O_P3>,N2>)
POP4X = DOT(P_O_P4>,N1>)
POP4Z = DOT(P_O_P4>,N2>)
POP5X = DOT(P_O_P5>,N1>)
POP5Z = DOT(P_O_P5>,N2>)
POP6X = DOT(P_O_P6>,N1>)
POP6Z = DOT(P_O_P6>,N2>)
POP7X = DOT(P_O_P7>,N1>)
POP7Z = DOT(P_O_P7>,N2>)
POP8X = DOT(P_O_P8>,N1>)
POP8Z = DOT(P_O_P8>,N2>)
POP9X = DOT(P_O_P9>,N1>)
POP9Z = DOT(P_O_P9>,N2>)
POP10X = DOT(P_O_P10>,N1>)
POP10Z = DOT(P_O_P10>,N2>)
POAOX = DOT(P_O_AO>,N1>)
POAOZ = DOT(P_O_AO>,N2>)
POBOX = DOT(P_O_BO>,N1>)
POBOZ = DOT(P_O_BO>,N2>)
POCOX = DOT(P_O_CO>,N1>)
POCOZ = DOT(P_O_CO>,N2>)
Appendix 3b
186
PODOX = DOT(P_O_DO>,N1>)
PODOZ = DOT(P_O_DO>,N2>)
POEOX = DOT(P_O_EO>,N1>)
POEOZ = DOT(P_O_EO>,N2>)
POFOX = DOT(P_O_FO>,N1>)
POFOZ = DOT(P_O_FO>,N2>)
POGOX = DOT(P_O_GO>,N1>)
POGOZ = DOT(P_O_GO>,N2>)
POCMX = DOT(P_O_CM>,N1>)
POCMZ = DOT(P_O_CM>,N2>)
%--------------------------------------------------------------------
% KINEMATICAL DIFFERENTIAL EQUATIONS
Q1' = U1
Q2' = U2
Q3' = U3
Q4' = U4
Q5' = U5
Q6' = U6
Q7' = U7
Q8' = U8
Q9' = U9
%--------------------------------------------------------------------
% ANGULAR VELOCITY & ACCELERATIONS
W_E_N> = U3*E3>
W_E_D> = U7*E3>
W_C_D> = U6*C3>
W_C_B> = U5*C3>
W_B_A> = U4*B3>
W_F_E> = U8*F3>
W_F_G> = U9*F3>
W_T_B> = 0>
ALF_E_N> = U3'*E3>
ALF_E_D> = U7'*E3>
ALF_C_D> = U6'*C3>
ALF_C_B> = U5'*C3>
ALF_B_A> = U4'*B3>
ALF_F_E> = U8'*F3>
ALF_F_G> = U9'*F3>
ALF_T_B> = 0>
%--------------------------------------------------------------------
% LINEAR VELOCITY
V_O_N> = 0>
V_P1_N> = DT(P_O_P1>,N)
V2PTS(N,A,P1,AO)
V2PTS(N,A,P1,P2)
V2PTS(N,B,P2,BO)
V2PTS(N,B,P2,P3)
V2PTS(N,T,P3,P4)
V2PTS(N,C,P3,CO)
V2PTS(N,C,P3,P5)
V2PTS(N,D,P5,DO)
V2PTS(N,D,P5,P6)
V2PTS(N,E,P6,EO)
V2PTS(N,E,P6,P7)
V2PTS(N,E,P6,P8)
V2PTS(N,F,P8,FO)
V2PTS(N,F,P8,P9)
V2PTS(N,G,P9,GO)
Appendix 3b
187
V2PTS(N,G,P9,P10)
V_CM_N> = DT(P_O_CM>,N)
VOP1X = DOT(V_P1_N>,N1>)
VOP1Z = DOT(V_P1_N>,N2>)
VOP2X = DOT(V_P2_N>,N1>)
VOP2Z = DOT(V_P2_N>,N2>)
VOP3X = DOT(V_P3_N>,N1>)
VOP3Z = DOT(V_P3_N>,N2>)
VOP4X = DOT(V_P4_N>,N1>)
VOP4Z = DOT(V_P4_N>,N2>)
VOP5X = DOT(V_P5_N>,N1>)
VOP5Z = DOT(V_P5_N>,N2>)
VOP6X = DOT(V_P6_N>,N1>)
VOP6Z = DOT(V_P6_N>,N2>)
VOP7X = DOT(V_P7_N>,N1>)
VOP7Z = DOT(V_P7_N>,N2>)
VOP8X = DOT(V_P8_N>,N1>)
VOP8Z = DOT(V_P8_N>,N2>)
VOP9X = DOT(V_P9_N>,N1>)
VOP9Z = DOT(V_P9_N>,N2>)
VOP10X = DOT(V_P10_N>,N1>)
VOP10Z = DOT(V_P10_N>,N2>)
VOAOX = DOT(V_AO_N>,N1>)
VOAOZ = DOT(V_AO_N>,N2>)
VOBOX = DOT(V_BO_N>,N1>)
VOBOZ = DOT(V_BO_N>,N2>)
VOCOX = DOT(V_CO_N>,N1>)
VOCOZ = DOT(V_CO_N>,N2>)
VODOX = DOT(V_DO_N>,N1>)
VODOZ = DOT(V_DO_N>,N2>)
VOEOX = DOT(V_EO_N>,N1>)
VOEOZ = DOT(V_EO_N>,N2>)
VOFOX = DOT(V_FO_N>,N1>)
VOFOZ = DOT(V_FO_N>,N2>)
VOGOX = DOT(V_GO_N>,N1>)
VOGOZ = DOT(V_GO_N>,N2>)
VOCMX = DOT(V_CM_N>,N1>)
VOCMZ = DOT(V_CM_N>,N2>)
%--------------------------------------------------------------------
% LINEAR ACCELERATION
A_O_N> = 0>
A_P1_N> = DT(V_P1_N>,N)
A_P2_N> = DT(V_P2_N>,N)
A_P3_N> = DT(V_P3_N>,N)
A_P4_N> = DT(V_P4_N>,N)
A_P5_N> = DT(V_P5_N>,N)
A_P6_N> = DT(V_P6_N>,N)
A_P7_N> = DT(V_P7_N>,N)
A_P8_N> = DT(V_P8_N>,N)
A_P9_N> = DT(V_P9_N>,N)
A_P10_N> = DT(V_P10_N>,N)
A_AO_N> = DT(V_AO_N>,N)
A_BO_N> = DT(V_BO_N>,N)
A_CO_N> = DT(V_CO_N>,N)
A_DO_N> = DT(V_DO_N>,N)
A_EO_N> = DT(V_EO_N>,N)
A_FO_N> = DT(V_FO_N>,N)
A_GO_N> = DT(V_GO_N>,N)
A_CM_N> = DT(V_CM_N>,N)
AOP1X = DOT(A_P1_N>,N1>)
AOP1Z = DOT(A_P1_N>,N2>)
Appendix 3b
188
AOP2X = DOT(A_P2_N>,N1>)
AOP2Z = DOT(A_P2_N>,N2>)
AOP4X = DOT(A_P4_N>,N1>)
AOP4Z = DOT(A_P4_N>,N2>)
AOCMX = DOT(A_CM_N>,N1>)
AOCMZ = DOT(A_CM_N>,N2>)
%--------------------------------------------------------------------
% ENERGY
KEA = KE(A)
KEB = KE(B)
KEC = KE(C)
KED = KE(D)
KEE = KE(E)
KEF = KE(F)
KEG = KE(G)
KECM = KE(A,B,C,D,E,F,G)
PEA = -MA*G*POAOZ
PEB = -MB*G*POBOZ
PEC = -MC*G*POCOZ
PED = -MD*G*PODOZ
PEE = -ME*G*POEOZ
PEF = -MF*G*POFOZ
PEG = -MG*G*POGOZ
PECM = -M*G*POCMZ
%--------------------------------------------------------------------
% ANGULAR & LINEAR MOMENTUM
AMOM> = MOMENTUM(ANGULAR,CM)
ANGMOM = DOT(AMOM>,N3>)
LMOM> = MOMENTUM(LINEAR)
HORMOM = DOT(LMOM>,N1>)
VERMOM = DOT(LMOM>,N2>)
%--------------------------------------------------------------------
% FORCES
GRAVITY(G*N2>)
FORCE(P1,RX1*N1> + RZ1*N2>)
FORCE(P2,RX2*N1> + RZ2*N2>)
FORCE(P4,RX3*N1> + RZ3*N2>)
TBAL = T^3 % CALL TORQUE SUBROUTINE TO OVERWRITE
TANK = T^3
TKNE = T^3
THIP = T^3
TSHO = T^3
TELB = T^3
TORQUE(B/A,TBAL*N3>)
TORQUE(C/B,TANK*N3>)
TORQUE(C/D,TKNE*N3>)
TORQUE(E/D,THIP*N3>)
TORQUE(F/E,TSHO*N3>)
TORQUE(F/G,TELB*N3>)
%--------------------------------------------------------------------
% EQUATIONS OF MOTION
ZERO = FR() + FRSTAR()
Appendix 3b
189
KANE ()
%--------------------------------------------------------------------
% INPUTS
INPUT TINITIAL=0.0,TFINAL=1.0,INTEGSTP=0.001,PRINTINT=100
INPUT ABSERR=1.0E-08,RELERR=1.0E-07
INPUT G=-9.806,THETA=32.543
INPUT MA=0.2762,MB=1.5878,MC=7.7935,MD=14.9787,ME=29.2589,&
MF=3.2917,MG=3.1395
INPUT Q1=0,Q2=0,Q3=90,Q4=0,Q5=0,Q6=0,Q7=0,Q8=0,&
U1=0,U2=5,U3=0,U4=0,U5=0,U6=0,U7=0,U8=0
INPUT L1=0.650,L2=0.0283,L3=0.1450,L4=0.0608,L5=0.0171,L6=0.048,&
L7=0.078,L8=0.4065,L9=0.1711,L10=0.371,L11=0.1648,L12=0.8650,&
L13=0.3827,L14=0.5650,L15=0.2450,L16=0.1118,&
L17=0.4255,L18=0.1618
INPUT IA=0.0002,IB=0.0033,IC=0.0996,ID=0.1786,IE=1.5544,&
IF=0.0184,IG=0.0408
%--------------------------------------------------------------------
% OUTPUTS
OUTPUT T,POP1X,POP1Z,POP2X,POP2Z,POP3X,POP3Z,POP4X,POP4Z,&
POP5X,POP5Z,POP6X,POP6Z,POP7X,POP7Z,POP8X,POP8Z,&
POP9X,POP9Z,POP10X,POP10Z,POCMX,POCMZ
OUTPUT T,VOP1X,VOP1Z,VOP2X,VOP2Z,VOP3X,VOP3Z,VOP4X,VOP4Z,&
VOP5X,VOP5Z,VOP6X,VOP6Z,VOP7X,VOP7Z,VOP8X,VOP8Z,&
VOP9X,VOP9Z,VOP10X,VOP10Z
OUTPUT T,POCMX,POCMZ,VOCMX,VOCMZ,AOCMX,AOCMZ
OUTPUT T,Q3,Q4,Q5,Q6,Q7,Q8,Q9
OUTPUT T,U3,U4,U5,U6,U7,U8,U9
OUTPUT T,U3',U4',U5',U6',U7',U8',Q9'
OUTPUT T,TBAL,TANK,TKNE,THIP,TSHO,TELB
OUTPUT T,RX1,RZ1,RX2,RZ2,RX3,RZ3
OUTPUT T,HORMOM,VERMOM,ANGMOM
OUTPUT T,KECM,KEA,KEB,KEC,KED,KEE,KEF,KEG
OUTPUT T,PECM,PEA,PEB,PEC,PED,PEE,PEF,PEG
%--------------------------------------------------------------------
% UNITS
UNITS T=S,[M,MA,MB,MC,MD,ME,MF,MG]=KG
UNITS [IA,IB,IC,ID,IE,IF,IG]=KGM^2
UNITS [Q3,Q4,Q5,Q6,Q7,Q8,Q9,THETA]=DEG
UNITS [U3,U4,U5,U6,U7,U8,U9']=RAD/S
UNITS [U3',U4',U5',U6',U7',U8',U9']=RAD/S^2
UNITS [Q1,Q2,L1,L2,L3,L4,L5,L6,L7,L8,L9,L10,L11]=M
UNITS [L12,L13,L14,L15,L16,L17,L18]=M
UNITS [POP1X,POP1Z,POP2X,POP2Z,POP3X,POP3Z,POP4X,POP4Z]=M
UNITS [POP5X,POP5Z,POP6X,POP6Z,POP7X,POP7Z,POP8X,POP8Z]=M
UNITS [POP9X,POP9Z,POP10X,POP10Z,POCMX,POCMZ]=M
UNITS [VOP1X,VOP1Z,VOP2X,VOP2Z,VOP3X,VOP3Z,VOP4X,VOP4Z]=M/S
UNITS [VOP5X,VOP5Z,VOP6X,VOP6Z,VOP7X,VOP7Z,VOP8X,VOP8Z]=M/S
UNITS [VOP9X,VOP9Z,VOP10X,VOP10Z,VOCMX,VOCMZ,U1,U2]=M/S
UNITS [U1',U2',AOCMX,AOCMZ,G]=M/S^2
UNITS [RX1,RX2,RX3,RZ1,RZ2,RZ3]=N
UNITS [TBAL,TANK,TKNE,THIP,TSHO,TELB]=NM
UNITS ANGMOM=KGM^2/S,[HORMOM,VERMOM]=KGM/S
UNITS [KECM,KEA,KEB,KEC,KED,KEE,KEF,KEG]=J
UNITS [PECM,PEA,PEB,PEC,PED,PEE,PEF,PEG]=J
%--------------------------------------------------------------------
SAVE TRAMPTQ.ALL
Appendix 3b
190
CODE DYNAMICS() TRAMPTQ.F, SUBS
%--------------------------------------------------------------------
% END END END END END END END END END END END END END
%--------------------------------------------------------------------
191
APPENDIX 4
Joint Torque Profiles from Angle-Driven
Matching Simulations
4a. Metatarsal-phalangeal joint
4b. Ankle joint
4c. Knee joint
4d. Hip joint
4e. Shoulder joint
4f. Elbow joint
Legend
S
F1
F2
F3
B1
B2
B3
Straight Jumping
Straight single forward somersault
Piked 1¾ forward somersault
Piked triffus
Piked single backward somersault
Straight single backward somersault
Piked double backward somersault
Appendix 4
192
4a.
4b.
4c.
-120
-100
-80
-60
-40
-20
0
20
0.0 0.2 0.4To
rqu
e N
m
Time s
-800
-600
-400
-200
0
200
400
0.0 0.2 0.4
Torq
ue
Nm
Time s
-300
-200
-100
0
100
200
0.0 0.2 0.4
Torq
ue
Nm
Time s
Appendix 4
193
4d.
4e.
4f.
-400
-300
-200
-100
0
100
200
300
400
0.0 0.2 0.4
Torq
ue
Nm
Time s
-150
-100
-50
0
50
100
150
0.0 0.2 0.4
Torq
ue
Nm
Time s
-150
-100
-50
0
50
100
0.0 0.2 0.4
Torq
ue
Nm
Time s
194
APPENDIX 5
Torque Generator Activation Parameters
from Strength Scaling Torque-Driven
Matching Simulations
5a. Straight front somersault, F1
5b. Piked 1 ¾ front somersault, F2
5c. Piked triffus, F3
Appendix 5
195
5a.
A0 A1 A2 TS1 TR1 I TR2
Ankle Plantar Flexion 0.049 1.000 0.330 -0.018 0.186 0.017 0.161
Ankle Dorsi Flexion 0.157 0.000 0.168 0.019 0.129 0.019 0.180
Knee Extension 0.102 1.000 0.192 0.017 0.109 0.097 0.145
Knee Flexion 0.307 0.047 0.475 -0.018 0.229 0.041 0.192
Hip Extension 0.043 1.000 0.250 -0.015 0.117 0.010 0.140
Hip Flexion 0.057 0.000 0.453 -0.020 0.130 0.009 0.176
Shoulder Flexion 0.297 0.792 0.115 -0.008 0.109 0.011 0.156
Shoulder Extension 0.341 0.170 0.916 -0.030 0.100 0.093 0.134
5b.
A0 A1 A2 TS1 TR1 I TR2
Ankle Plantar Flexion 0.142 1.000 0.544 0.004 0.194 0.000 0.125
Ankle Dorsi Flexion 0.458 0.000 0.419 -0.019 0.145 0.004 0.103
Knee Extension 0.142 1.000 0.292 0.020 0.139 0.063 0.189
Knee Flexion 0.427 0.224 0.183 -0.003 0.207 0.057 0.292
Hip Extension 0.059 1.000 0.266 -0.029 0.154 0.014 0.180
Hip Flexion 0.078 0.000 0.831 -0.009 0.128 0.007 0.155
Shoulder Flexion 0.201 0.902 0.075 -0.002 0.102 0.014 0.127
Shoulder Extension 0.232 0.079 0.741 -0.040 0.145 0.012 0.154
Appendix 5
196
5c.
A0 A1 A2 TS1 TR1 I TR2
Ankle Plantar Flexion 0.158 1.000 0.557 0.007 0.184 0.020 0.179
Ankle Dorsi Flexion 0.508 0.000 0.413 0.006 0.121 0.002 0.127
Knee Extension 0.158 1.000 0.202 -0.014 0.106 0.071 0.130
Knee Flexion 0.473 0.047 0.776 0.015 0.257 0.136 0.202
Hip Extension 0.066 1.000 0.001 0.010 0.116 0.015 0.102
Hip Flexion 0.089 0.000 0.958 0.000 0.150 0.100 0.103
Shoulder Flexion 0.191 0.503 0.144 -0.036 0.140 0.024 0.100
Shoulder Extension 0.220 0.156 0.615 -0.040 0.104 0.201 0.176
197
APPENDIX 6
Torque Generator Activation Parameters
from Fixed Strength Torque-Driven
Matching Simulations
6a. Straight front somersault, F1
6b. Piked 1 ¾ front somersault, F2
6c. Piked triffus, F3
Appendix 6
198
6a.
A0 A1 A2 TS1 TR1 I TR2
Ankle Plantar Flexion 0.024 0.591 0.577 -0.016 0.181 0.009 0.129
Ankle Dorsi Flexion 0.090 0.011 0.577 0.016 0.127 0.003 0.205
Knee Extension 0.081 0.810 0.062 0.018 0.109 0.074 0.130
Knee Flexion 0.234 0.002 0.976 -0.012 0.235 0.116 0.208
Hip Extension 0.038 0.926 0.186 -0.015 0.117 0.003 0.169
Hip Flexion 0.051 0.004 0.267 -0.017 0.113 0.023 0.134
Shoulder Flexion 0.204 0.549 0.012 -0.009 0.112 0.007 0.154
Shoulder Extension 0.226 0.119 0.905 -0.030 0.101 0.102 0.214
6b.
A0 A1 A2 TS1 TR1 I TR2
Ankle Plantar Flexion 0.123 0.862 0.846 0.004 0.189 0.010 0.132
Ankle Dorsi Flexion 0.379 0.048 0.682 -0.018 0.141 0.005 0.127
Knee Extension 0.069 0.456 0.724 0.019 0.139 0.092 0.189
Knee Flexion 0.205 0.080 0.200 -0.006 0.220 0.129 0.249
Hip Extension 0.061 0.996 0.263 -0.030 0.158 0.011 0.186
Hip Flexion 0.069 0.004 0.799 -0.008 0.125 0.016 0.117
Shoulder Flexion 0.170 0.746 0.027 -0.002 0.102 0.030 0.145
Shoulder Extension 0.192 0.067 0.621 -0.040 0.146 0.001 0.143
Appendix 6
199
6c.
A0 A1 A2 TS1 TR1 I TR2
Ankle Plantar Flexion 0.042 0.993 0.024 0.004 0.184 0.011 0.175
Ankle Dorsi Flexion 0.069 0.263 0.874 -0.012 0.397 0.005 0.106
Knee Extension 0.220 0.857 0.537 -0.013 0.110 0.028 0.184
Knee Flexion 0.435 0.089 0.377 0.010 0.254 0.379 0.130
Hip Extension 0.043 0.932 0.004 0.011 0.121 0.018 0.116
Hip Flexion 0.069 0.228 0.339 0.016 0.449 0.150 0.182
Shoulder Flexion 0.202 0.515 0.037 -0.036 0.137 0.002 0.138
Shoulder Extension 0.224 0.146 0.316 -0.034 0.110 0.313 0.215
200
APPENDIX 7
Torque Generator Activation Parameters
from Optimisations
7a. Height in a single straight somersault
7b. Height in a 1¾ piked somersault
7c. Height in a piked triffus
7d. Rotation using half the length of the jump zone
7e. Rotation using the full length of the jump zone
Appendix 7
201
7a.
A0 A1 A2 TS1 TR1 I TR2
Ankle Plantar Flexion 0.035 0.687 0.652 0.164 0.292 0.016 0.149
Ankle Dorsi Flexion 0.090 0.002 0.468 0.022 0.117 0.014 0.210
Knee Extension 0.105 0.904 0.292 0.021 0.117 0.085 0.164
Knee Flexion 0.237 0.015 0.930 -0.007 0.235 0.035 0.227
Hip Extension 0.070 0.899 0.129 -0.013 0.112 0.020 0.162
Hip Flexion 0.053 0.004 0.222 -0.011 0.119 0.032 0.124
Shoulder Flexion 0.242 0.466 0.090 -0.016 0.114 0.003 0.146
Shoulder Extension 0.191 0.119 0.877 -0.023 0.102 0.108 0.245
7b.
A0 A1 A2 TS1 TR1 I TR2
Ankle Plantar Flexion 0.105 0.909 0.935 0.194 0.796 0.001 0.136
Ankle Dorsi Flexion 0.300 0.040 0.582 -0.010 0.151 0.015 0.137
Knee Extension 0.066 0.448 0.796 0.019 0.129 0.086 0.194
Knee Flexion 0.206 0.079 0.142 -0.009 0.227 0.125 0.253
Hip Extension 0.061 0.994 0.363 -0.030 0.154 0.021 0.196
Hip Flexion 0.068 0.004 0.700 -0.006 0.134 0.026 0.127
Shoulder Flexion 0.173 0.771 0.127 -0.002 0.103 0.021 0.135
Shoulder Extension 0.192 0.071 0.585 -0.039 0.144 0.004 0.135
Appendix 7
202
7c.
A0 A1 A2 TS1 TR1 I TR2
Ankle Plantar Flexion 0.035 0.999 0.010 0.188 0.637 0.018 0.168
Ankle Dorsi Flexion 0.128 0.165 0.870 -0.010 0.392 0.005 0.102
Knee Extension 0.280 0.869 0.637 -0.007 0.112 0.024 0.188
Knee Flexion 0.468 0.018 0.310 0.012 0.246 0.384 0.135
Hip Extension 0.045 0.997 0.069 0.011 0.111 0.008 0.122
Hip Flexion 0.070 0.149 0.376 0.013 0.450 0.145 0.175
Shoulder Flexion 0.177 0.500 0.130 -0.038 0.132 0.011 0.138
Shoulder Extension 0.218 0.146 0.406 -0.036 0.119 0.319 0.223
7d.
A0 A1 A2 TS1 TR1 I TR2
Ankle Plantar Flexion 0.031 0.999 0.000 0.193 0.637 0.002 0.165
Ankle Dorsi Flexion 0.082 0.288 0.874 -0.009 0.395 0.012 0.107
Knee Extension 0.286 0.820 0.637 -0.014 0.113 0.037 0.193
Knee Flexion 0.430 0.013 0.354 0.006 0.249 0.378 0.123
Hip Extension 0.049 0.931 0.001 0.014 0.112 0.008 0.107
Hip Flexion 0.067 0.316 0.379 0.013 0.445 0.152 0.185
Shoulder Flexion 0.222 0.508 0.117 -0.036 0.140 0.009 0.130
Shoulder Extension 0.203 0.169 0.382 -0.025 0.113 0.305 0.210
Appendix 7
203
7e.
A0 A1 A2 TS1 TR1 I TR2
Ankle Plantar Flexion 0.119 0.990 0.002 0.181 0.602 0.020 0.165
Ankle Dorsi Flexion 0.169 0.321 0.864 -0.002 0.397 0.003 0.108
Knee Extension 0.235 0.924 0.602 -0.016 0.111 0.034 0.181
Knee Flexion 0.354 0.112 0.436 0.013 0.252 0.370 0.128
Hip Extension 0.027 0.840 0.001 0.005 0.114 0.008 0.106
Hip Flexion 0.064 0.320 0.376 0.019 0.440 0.145 0.177
Shoulder Flexion 0.186 0.493 0.083 -0.040 0.134 0.004 0.134
Shoulder Extension 0.225 0.145 0.232 -0.029 0.102 0.310 0.215